A probability distribution is an idealized frequency distribution. In statistics, a frequency distribution represents the number of occurrences of different outcomes in a dataset. It shows how often each different value appears within a dataset.
Probability distribution represents an abstract representation of the frequency distribution. While a frequency distribution pertains to a particular sample or dataset, detailing how often each potential value of a variable appears within it, the occurrence of each value in the sample is dictated by its probability.
A probability distribution, not only shows the frequencies of different outcomes but also assigns probabilities to each outcome. These probabilities indicate the likelihood of each outcome occurring.
In this article, we will learn what is probability distribution, types of probability distribution, probability distribution function, and formulas.
Table of Content
- What is Probability Distribution?
- Random Variables
- Types of Random Variables in Probability Distribution
- Probability Distribution of a Random Variable
- Probability Distribution Formulas
- Expectation (Mean) and Variance of a Random Variable
- Different Types of Probability Distributions
- Discrete Probability Distributions
- Cumulative Probability Distribution
- Probability Distribution Function
- Probability Distribution Table
- Prior Probability
- Posterior Probability
- Solved Questions on Probability Distribution
What is Probability Distribution?
In Probability Distribution, A Random Variable’s outcome is uncertain. Here, the outcome’s observation is known as Realization. It is a Function that maps Sample Space into a Real number space, known as State Space. They can be Discrete or Continuous.Â
Probability Distribution Definition
The probability Distribution of a Random Variable (X) shows how the Probabilities of the events are distributed over different values of the Random Variable. When all values of a Random Variable are aligned on a graph, the values of its probabilities generate a shape. The Probability distribution has several properties (for example: Expected value and Variance) that can be measured. It should be kept in mind that the Probability of a favorable outcome is always greater than zero and the sum of all the probabilities of all the events is equal to 1.
Probability Distribution is basically the set of all possible outcomes of any random experiment or event.
Also Read: Events in Probability
Random Variables
Random Variable is a real-valued function whose domain is the sample space of the random experiment. It is represented as X(sample space) = Real number.Â
We need to learn the concept of Random Variables because sometimes we are just only interested in the probability of the event but also in the number of events associated with the random experiment. The importance of random variables can be better understood by the following example:
Why do we need Random Variables?
Let’s take an example of the coin flips. We’ll start with flipping a coin and finding out the probability. We’ll use H for ‘heads’ and T for ‘tails’.Â
So now we flip our coin 3 times, and we want to answer some questions.Â
- What is the probability of getting exactly 3 heads?
- What is the probability of getting less than 3 heads?
- What is the probability of getting more than 1 head?
Then our general way of writing would be:
- P(Probability of getting exactly 3 heads when we flip a coin 3 times)
- P(Probability of getting less than 3 heads when we flip a coin 3 times)
- P(Probability of getting more than 1 head when we flip a coin 3 times)Â
In a different scenario, suppose we are tossing two dice, and we are interested in knowing the probability of getting two numbers such that their sum is 6.Â
So, in both of these cases, we first need to know the number of times the desired event is obtained i.e. Random Variable X in sample space which would be then further used to compute the Probability P(X) of the event. Hence, Random Variables come to our rescue. First, let’s define what is random variable mathematically.Â
Random Variable Definition
Random Variable is a function that associates a real number with an event. It means assigning a value (real number) to every possible outcome. In more mathematical terms, it is a function from the sample space â„¦ to the real numbers. We can choose our random variable according to our needs.Â
A random variable is a real valued function whose domain is the sample space of a random experiment
To understand this concept in a lucid manner, let us consider the experiment of tossing a coin two times in succession.
The sample space of the experiment is S = {HH, HT, TH, TT}. Let’s define a random variable to count events of head or tails according to our need, let X is a random variable that denotes the number of heads obtained. For each outcome, its values are as given below:
X(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0.
More than one random variable can be defined in the same sample space. For example, let Y is a random variable denoting the number of heads minus the number of tails for each outcome of the above sample space S.
Â Y(HH) = 2-0 = 2; Y (HT) = 1-1 = 0; Y (TH) = 1-1= 0; Y (TT) = 0-2 = â€“ 2
Thus, X and Y are two different random variables defined on the same sample.
Note: More than one event can map to same value of random variable.Â
Types of Random Variables in Probability Distribution
There are following two types ofÂ
- Discrete Random Variables
- Continuous Random Variables
Discrete Random Variables in Probability Distribution
A Discrete Random Variable can only take a finite number of values. To further understand this, let’s see some examples of discrete random variables:
- X = {sum of the outcomes when two dice are rolled}. Here, X can only take values like {2, 3, 4, 5, 6….10, 11, 12}.
- X = {Number of Heads in 100 coin tosses}. Here, X can take only integer values from [0,100].
Continuous Random Variable in Probability Distribution
A Continuous Random Variable can take infinite values in a continuous domain. Let’s see an example of a dart game.Â
Suppose, we have a dart game in which we throw a dart where the dart can fall anywhere between [-1,1] on the x-axis. So if we define our random variable as the x-coordinate of the position of the dart, X can take any value from [-1,1]. There are infinitely many possible values that X can take. (X = {0.1, 0.001, 0.01, 1,2, 2.112121 …. and so on}. Â Â
Probability Distribution of a Random Variable
Now the question comes, how to describe the behavior of a random variable?Â
Suppose that our Random Variable only takes finite values, like x_{1}, x_{2}, x_{3} …. and x_{n}. ie. the range ofÂ X is the set of n values is {x_{1}, x_{2}, x_{3} …. and x_{n}}.
Thus, the behavior of X is completely described by giving probabilities for all the values of the random variable X
Event | Probability |
---|---|
x_{1} | P(X = x_{1}) |
x_{2} | P(X = x_{2}) |
x_{3} | P(X = x_{3}) |
The Probability Function of a discrete random variableÂ X is the functionÂ p(x) satisfying
P(x) = P(X = x)
Let’s look at an example:Â
Example: We draw two cards successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution of finding aces.
Answer:Â
Let’s define a random variable “X”, which means number of aces. So since we are only drawing two cards from the deck, X can only take three values: 0, 1 and 2. Â We also know that, we are drawing cards with replacement which means that the two draws can be considered an independent experiments.Â
P(X = 0) = P(both cards are non-aces)Â
Â Â Â Â Â Â Â Â = P(non-ace) x P(non-ace)Â
Â Â Â Â Â Â Â Â =Â [Tex]\frac{48}{52} \times \frac{48}{52} = \frac {144}{169} [/Tex]
P(X = 1) = P(one of the cards in ace)Â
Â Â Â Â Â Â Â Â = P(non-ace and then ace) + P(ace and then non-ace)
Â Â Â Â Â Â Â Â = P(non-ace) x P(ace) + P(ace) x P(non-ace)
Â Â Â Â Â Â Â Â =Â [Tex]\frac{48}{52} \times \frac{4}{52} Â + \frac{4}{52} \times \frac{48}{52} = \frac{24}{169} [/Tex]
P(X = 2) = P(Both the cards are aces)Â
Â Â Â Â Â Â Â Â = P(ace) x P(ace)
Â Â Â Â Â Â Â Â =Â [Tex]\frac{4}{52} \times \frac{4}{52} = \frac{1}{169} [/Tex]
Now we have probabilities for each value of random variable. Since it is discrete, we can make a table to represent this distribution. The table is given below.Â
XÂ | 0 | 1 | 2 |
---|---|---|---|
P(X=x) | [Tex]\frac{144}{169} [/Tex] | [Tex]\frac{24}{169} [/Tex] | [Tex]\frac{1}{169} [/Tex] |
It should be noted here that each value of P(X=x) is greater than zero and the sum of all P(X=x) is equal to 1.
Probability Distribution Formulas
The various formulas under Probability Distribution are tabulated below:
Types of Distribution | Formula |
---|---|
Binomial Distribution | P(X) =Â ^{n}C_{x}a^{x}b^{n-x} Where a = probability of success b=probability of failure n= number of trials x=random variable denoting success |
Cumulative Distribution Function | F_{x}(x) = [Tex]\int_{-âˆž}^{x} [/Tex]f(x)(t)dt |
Discrete Probability Distribution | P(x) = n!/ r!(n-r)! . p^{r}(1-p)^{n-r} P(x) = C(n,r) . p^{r}(1-p)^{n-r} |
Expectation (Mean) and Variance of a Random Variable
Suppose we have a probability experiment we are performing, and we have defined some random variable(R.V.) according to our needs( like we did in some previous examples). Now, each time experiment is performed, our R.V. takes on a different value. But we want to know that if we keep on doing the experiment a thousand times or an infinite number of times, what will be the average value of the random variable?
Expectation
The mean, expected value, or expectation of a random variable X is written as E(X) orÂ [Tex]\mu_{\textbf{X}}. [/Tex]Â If we observe N random values of X, then the mean of the N values will be approximately equal to E(X) for large N.Â
For a random variable X which takes on values x_{1}, x_{2, }x_{3 … }x_{n} with probabilities Â p_{1}, p_{2}, p_{3} … p_{n}. Expectation of X is defined as,Â
[Tex]\sum_{i=1}^{N} x_{i}p_{i} [/Tex]
i.e it is weighted average of all values which X can take, weighted by the probability of each value.Â
To see it more intuitively, let’s take a look at this graph below,Â
Now in the above figure, we can see both the Random Variables have the almost same ‘mean’, but does that mean that they are equal? No. To fully describe the properties/behavior of a random variable, we need something more, right?Â
We need to look at the dispersion of the probability distribution, one of them is concentrated, but the other is very spread out near a single value. So we need a metric to measure the dispersion in the graph.Â
Variance
In Statistics, we have studied that the variance is a measure of the spread or scatter in the data. Likewise, the variability or spread in the values of a random variable may be measured by variance.
For a random variable X which takes on values x_{1}, x_{2}, x_{3} … x_{n} with probabilities Â p_{1}, p_{2}, p_{3} … p_{n} and the expectation is Â E[X]Â
The variance of X or Var(X) is denoted by,Â [Tex]E[X – u]^{2} = \sum (x_{i}-\mu)^{2}p_{x_{i}} = E[X^{2}] – (E[X])^{2} [/Tex]
Let’s calculate the mean and variance of a random variable probability distribution through an example:
Example: Find the variance and mean of the number obtained on a throw of an unbiased die.
Answer:Â
We know that the sample space of this experiment is {1,2,3,4,5,6}Â
Let’s define our random variable X, which represents the number obtained on a throw.Â
So, the probabilities of the values which our random variable can take are,Â
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) =Â [Tex]\frac{1}{6} [/Tex]
Therefore, the probability distribution of the random variable is,Â
X 1 2 3 4 5 6 Probabilities [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6 } [/Tex] [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6} [/Tex] E[X] =Â [Tex]\sum p_{x_{i}}x_{i} \\ \hspace{0.9cm} = 1 \times \frac{1}{6} + 2 \times \frac{1}{6} + 3 \times \frac{1}{6} + 4 \times \frac{1}{6} + 5 \times \frac{1}{6} + 6 \times \frac{1}{6} \\ \hspace{0.9cm} = \frac{21}{6} [/Tex]
Also, E[X^{2}] =Â [Tex]1^{2} \times \frac{1}{6} + 2^{2}\times\frac{1}{6} + 3^{2}\times\frac{1}{6} + 4^{2}\times\frac{1}{6} + 5^{2}\times\frac{1}{6} + 6^{2}\times\frac{1}{6} \\ \hspace{0.9cm} = \frac{91}{6} \\ [/Tex]
Thus, Var(X) = E[X^{2}] – (E[X])^{2}
Â Â Â Â Â Â Â Â Â Â Â =Â [Tex](\frac{91}{6}) – (\frac{21}{6})^{2} = \frac{91}{6} – \frac{441}{36} = \frac{35}{12} [/Tex]
So, therefore mean isÂ [Tex]\frac{21}{6} [/Tex]and variance isÂ [Tex]\frac{35}{12} [/Tex]
Different Types of Probability Distributions
We have seen what Probability Distributions are, now we will see different types of Probability Distributions. The Probability Distribution’s type is determined by the type of random variable. There are two types of Probability Distributions:Â
- Discrete Probability Distributions for discrete variables
- Cumulative Probability Distribution for continuous variables
We will study in detail two types of discrete probability distributions, others are out of scope at class 12.Â
Discrete Probability Distributions
Discrete Probability Functions also called Binomial Distribution assume a discrete number of values. For example, coin tosses and counts of events are discrete functions. These are discrete distributions because there are no in-between values. We can either have heads or tails in a coin toss.
For discrete probability distribution functions, each possible value has a non-zero probability. Moreover, the sum of all the values of probabilities must be one. For example, the probability of rolling a specific number on a die is 1/6. The total probability for all six values equals one. When we roll a die, we only get either one of these values.
Bernoulli Trials and Binomial Distributions
When we perform a random experiment either we get the desired event or we don’t. If we get the desired event then we call it a success and if we don’t it is a failure. Let’s say in the coin-tossing experiment if the occurrence of the head is considered a success, then the occurrence of the tail is a failure.
Each time we toss a coin or roll a die or perform any other experiment, we call it a trial. Now we know that in our experiments coin-tossing trial, the outcome of any trial is independent of the outcome of any other trial. In each of such trials, the probability of success or failure remains constant. Such independent trials that have only two outcomes usually referred to as â€˜successâ€™ or â€˜failureâ€™ are called Bernoulli Trials.
Definition:
Trials of the random experiment are known as Bernoulli Trials, if they are satisfying below given conditions :
- Finite number of trials are required.
- All trials must be independent.
- Every trial has two outcomes : success or failure.
- Probability of success remains Â same in every trial.
Let’s take the example of an experiment in which we throw a die; throwing a die 50 times can be considered as a case of 50 Bernoulli trials, where the result of each trial is either success(let’s assume that getting an even number is a success) or failure( similarly, getting an odd number is failure) and the probability of success (p) is the same for all 50 throws. Obviously, the successive throws of the die are independent trials. If the die is fair and has six numbers 1 to 6 written on six faces, then p = 1/2 is the probability of success, and q = 1 â€“ p =1/2 is the probability of failure.
Example: An urn contains 8 red balls and 10 black balls. We draw six balls from the urn successively. You have to tell whether or not the trials of drawing balls are Bernoulli trials when after each draw, the ball drawn is:
- replaced
- not replaced in the urn.
Answer:
- We know that the number of trials are finite. When drawing is done with replacement, probability of success (say, red ball) is p =8/18 which will be same for all of the six trials. So, drawing of balls with replacements are Bernoulli trials.
- If Â drawing is done without replacement, Â probability of success (i.e., red ball) in the first trial is 8/18 , in 2nd trial is 7/17 if Â first ball drawn is red or, 10/18 if Â first ball drawn is black, and so on. Clearly, probabilities of success are not same for all the trials, Therefore, the trials are not Bernoulli trials.
Binomial Distribution
It is a random variable that represents the number of successes in “N” successive independent trials of Bernoulli’s experiment. It is used in a plethora of instances including the number of heads in “N” coin flips, and so on.Â
Let P and Q denote the success and failure of the Bernoulli Trial respectively. Let’s assume we are interested in finding different ways in which we have 1 success in all six trials.
Clearly, six cases are available as listed below:
PQQQQQ, QPQQQQ, QQPQQQ, QQQPQQ, QQQQPQ, QQQQQP
Likewise, 2 successes and 4 failures will showÂ [Tex]\frac{6!}{4! 2!}Â [/Tex]Â combinations thus making it difficult to list so many combinations. Henceforth, calculating probabilities of 0, 1, 2,…, n number of successes can be long and time-consuming. To avoid such lengthy calculations along with a listing of all possible cases, for probabilities of the number of successes in n-Bernoulli’s trials, a formula is made which is given as:
If Y is a Binomial Random Variable, we denote this Yâˆ¼ Bin(n, p), where p is the probability of success in a given trial, q is the probability of failure, Let ‘n’ be the total number of trials, and ‘x’ be the number of successes, the Probability Function P(Y) for Binomial Distribution is given as:
P(Y) = ^{n}C_{x} q^{nâ€“x}p^{x}Â
where x = 0,1,2…n
Example: When a fair coin is tossed 10 times, find the probability of getting:
- Exactly Six Heads
- At least Six Heads
Answer:Â
Every coin tossed can be considered as the Bernoulli trial. Suppose X is the number of heads in this experiment:Â
We already know, n = 10
Â Â Â Â Â Â Â Â Â Â Â Â Â Â p = 1/2
So, P(X = x) = ^{n}C_{x} p^{n-x} (1-p)^{x }, x= 0,1,2,3,….n
P(X = x) = ^{10}C_{x}p^{10-x}(1-p)^{x} Â
When x = 6,Â
(i) P(x = 6) = ^{10}C_{6} p^{4} (1-p)^{6}
^{Â Â Â Â Â Â Â Â Â Â }=Â [Tex]\frac{10!}{6!4!}(\frac{1}{2})^{6}(\frac{1}{2})^{4}\\ \hspace{0.4cm} = \frac{7\times8\times9\times10}{2\times3\times4}\times\frac{1}{64}\times\frac{1}{16} \\ \hspace{0.4cm} = \frac{105}{512} [/Tex]
(ii) P(at least 6 heads) = P(X >= 6) = P(X = 6) + P(X=7) + P(X=8)+ P(X=9) + P(X=10)Â
= ^{10}C_{6} p^{4 }(1-p)^{6 }+ ^{10}C_{7 }p^{3 }(1-p)^{7} + ^{10}C_{8 }p^{2 }(1-p)^{8 }+ ^{10}C_{9 }p^{1}(1-p)^{9} + ^{10}C_{10 }(1-p)^{10 }= Â
[Tex]\frac{10!}{6!4!}(\frac{1}{2})^{10} + \frac{10!}{7!3!}(\frac{1}{2})^{10} + \frac{10!}{8!2!}(\frac{1}{2})^{10} + \frac{10!}{9!1!}(\frac{1}{2})^{10} + \frac{10!}{10!}(\frac{1}{2})^{10}\\ \hspace{0.5cm} = (\frac{10!}{6!4!} + \frac{10!}{7!3!}+ \frac{10!}{8!2!} + \frac{10!}{9!1!}+ \frac{10!}{10!})(\frac{1}{2})^{10} \\ \hspace{0.5cm} = \frac{193}{512} [/Tex]
Negative Binomial Distribution
In a random experiment of discrete range, it is not necessary that we get success in every trial. If we perform ‘n’ number of trials and get success ‘r’ times where n>r, then our failure will be (n-r) times. The probability distribution of failure in this case will be called negative binomial distribution. For example, if we consider getting 6 in the die is success and we want 6 one time, but 6 is not obtained in the first trial then we keep throwing the die until we get 6. Suppose we get 6 in the sixth trial then the first 5 trials will be failures and if we plot the probability distribution of these failures then the plot so obtained will be called as negative binomial distribution.
Poisson Probability Distribution
The Probability Distribution of the frequency of occurrence of an event over a specific period is called Poisson Distribution. It tells how many times the event occurred over a specific period. It basically counts the number of successes and takes a value of the whole number i.e. (0,1,2…). It is expressed as
f(x; Î») = P(X=x) = (Î»^{x}e^{-Î»})/x!
where,Â
- x is number of times event occurred
- e = 2.718…
- Î» is mean value
Binomial Distribution Examples
Binomial Distribution is used for the outcomes that are discrete in nature. Some of the examples where Binomial Distribution can be used are mentioned below:
- To find the number of good and defective items produced by a factory.
- To find the number of girls and boys studying in a school.
- To find out the negative or positive feedback on something
Cumulative Probability Distribution
The Cumulative Probability Distribution for continuous variables is a function that gives the probability that a random variable takes on a value less than or equal to a specified point. It’s denoted as F(x), where x represents a specific value of the random variable. For continuous variables, F(x) is found by integrating the probability density function (pdf) from negative infinity to x. The function ranges from 0 to 1, is non-decreasing, and right-continuous. It’s essential for computing probabilities, determining percentiles, and understanding the behavior of continuous random variables in various fields.
Cumulative Probability Distribution takes value in a continuous range; for example, the range may consist of a set of real numbers. In this case, Cumulative Probability Distribution will take any value from the continuum of real numbers unlike the discrete or some finite value taken in the case of Discrete Probability distribution. Cumulative Probability Distribution is of two types, Continuous Uniform Distribution, and Normal Distribution.
Continuous Uniform Distribution
Continuous Uniform Distribution is described by a density function that is flat and assumes value in a closed interval let’s say [P, Q] such that the probability is uniform in this closed interval. It is represented as f(x; P, Q)
f(x; P, Q) = 1/(Q-P) for Pâ‰¤xâ‰¤Q
f(x; P, Q) = 0; elsewhere
Normal Distribution
Normal Distribution of continuous random variables results in a bell-shaped curve. It is often referred to as Gaussian Distribution on the name of Karl Friedrich Gauss who derived its equation. This curve is frequently used by the meteorological department for rainfall studies. The Normal Distribution of random variable X is given by
n(x; Î¼, Ïƒ) = {1/(âˆš2Ï€)Ïƒ}e^{(-1/2Ïƒ^2)(x-Î¼)^2} for -âˆž<x<âˆžÂ
whereÂ
- Î¼ is mean
- Ïƒ is variance
Normal Distribution Examples
The Normal Distribution Curve can be used to show the distribution of natural events very well. Over the period it has become a favorite choice of statisticians to study natural events. Some of the examples where the Normal Distribution Curve can be used are mentioned below
- Salary of Working Class
- Life Expectancy of human in a Country
- Heights of Male or Female
- The IQ Level of children
- Expenditure of households
Probability Distribution Function
Probability Distribution Function is defined as the function that is used to express the distribution of a probability. Different types of probability, they are expressed differently. These functions are also used for Probability Density Functions for different variables.
For Normal Distribution, the Probability Distribution Function for Random Variable X is given by F_{x}(x) = P(X â‰¤ x) where X is the Random variable and P is the Probability.
The Cumulative Probability Distribution for closed interval aâ‡¢b is given as P(a < X â‰¤ b) = F_{x}(b) – F_{x}(a).Â
In terms of integrals, the cumulative probability function is given asÂ [Tex]F_{x}(x) = \int_{-\infty }^{x}f_{x}(t)dt [/Tex]
For Random Variable X = p, the Cumulative Probability function is given asÂ [Tex]P(X = p) = F_{x}(p) – \lim_{x\rightarrow p}f_{x}(t) [/Tex]
Binomial Probability Distribution gives some exact values. It is often called as Probability Mass Function. For a Random Variable X and Space S where X: Sâ‡¢A where A belongs to Random Discrete Variable R, X can be defined as f_{x}(x) = P_{r}(X = x) = P({s âˆˆ S: X(s) = x}).
Probability Distribution Table
The random variables and their corresponding probability is tabulated then it is called Probability Distribution Table. The following table represents a Probability Distribution Table:
X | X_{1} | X_{2} | X_{3} | X_{4} | …. | X_{n} |
---|---|---|---|---|---|---|
P(X) | P_{1} | P_{2} | P_{3} | P_{4} | …. | P_{n} |
It should be noted that the sum of all probabilities is equal to 1.
Prior Probability
Prior Probability as the name suggests refers to assigning the probability of an event before the happening of a dependent event that makes us make changes in the Prior Probability. Let’s say we assign Probability P(A) to event A before taking into account that event B has also happened. After B has happened we need to revise P(A) using Baye’s Theorem. Hence, here P(A) is the Prior Probability. If we predict that a particular observation will fall into a particular category before collecting all the observations, then this is also called Prior Probability.
Posterior Probability
After the Prior Probability has been assigned and new information is obtained then the Prior Probability is modified by taking into account the newly obtained information using Baye’s Formula. This revised probability is called Posterior Probability. Hence, we can say that Posterior Probability is a conditional probability obtained by revising the Prior Probability.
Chi-square distribution
Chi-square distribution is a probability distribution that arises in statistics, particularly in hypothesis testing and confidence interval estimation. It is characterized by its degrees of freedom, which determine its shape. The distribution is positively skewed and only takes non-negative values. The Chi-square distribution is widely used in inferential statistics for testing the independence of variables in contingency tables, assessing goodness of fit, and estimating population variances.
Chi-square table
Below is a Chi-square table showing critical values for selected degrees of freedom and levels of significance:
Degrees of Freedom (df) | 0.01 | 0.05 | 0.10 |
---|---|---|---|
1 | 6.63 | 3.84 | 2.71 |
2 | 9.21 | 5.99 | 4.61 |
3 | 11.34 | 7.81 | 6.25 |
4 | 13.28 | 9.49 | 7.78 |
5 | 15.09 | 11.07 | 9.24 |
6 | 16.81 | 12.59 | 10.64 |
7 | 18.48 | 14.07 | 12.02 |
8 | 20.09 | 15.51 | 13.36 |
9 | 21.67 | 16.92 | 14.68 |
10 | 23.21 | 18.31 | 15.99 |
This table provides critical values for the Chi-square distribution at various levels of significance (0.01, 0.05, and 0.10) and degrees of freedom (from 1 to 10). Critical values from the Chi-square table are commonly used in hypothesis testing to determine whether observed frequencies in a contingency table differ significantly from expected frequencies.
t Distribution
The t distribution, also known as the Student’s t-distribution, is a probability distribution that is similar to the standard normal distribution but has heavier tails. It is commonly used in hypothesis testing and constructing confidence intervals when the sample size is small or the population standard deviation is unknown. The shape of the t distribution depends on the sample size, and as the sample size increases, the t distribution approaches the standard normal distribution.
t Table
Below is a t-table showing critical values for selected degrees of freedom (df) and levels of significance:
Degrees of Freedom (df) | 0.01 | 0.05 | 0.10 |
---|---|---|---|
1 | 12.706 | 6.314 | 3.078 |
2 | 4.303 | 2.920 | 1.886 |
3 | 3.182 | 2.353 | 1.638 |
4 | 2.776 | 2.132 | 1.533 |
5 | 2.571 | 2.015 | 1.476 |
6 | 2.447 | 1.943 | 1.440 |
7 | 2.365 | 1.895 | 1.415 |
8 | 2.306 | 1.860 | 1.397 |
9 | 2.262 | 1.833 | 1.383 |
10 | 2.228 | 1.812 | 1.372 |
This table provides critical values for the t-distribution at various levels of significance (0.01, 0.05, and 0.10) and degrees of freedom (from 1 to 10). Critical values from the t-table are commonly used in hypothesis testing to determine whether sample means significantly differ from population means when the population standard deviation is unknown and sample sizes are small.
Solved Questions on Probability Distribution
Question 1: A box contains 4 blue balls and 3 green balls. Find the probability distribution of the number of green balls in a random draw of 3 balls.
Solution:
Given that the total number of balls is 7 out of which 3 have to be drawn at random. On drawing 3 balls the possibilities are all 3 are green, only 2 is green, only 1 is green, and no green. Hence X = 0, 1, 2, 3.
P(No ball is green) = P(X = 0) = ^{4}C_{3}/^{7}C_{3} = 4/35
P(1 ball is green) = P(X = 1) = ^{3}C_{1} Ã— ^{4}C_{2}/^{7}C_{3} = 18/35
P(2 balls are green) = P(X = 2) = ^{3}C_{2} Ã— ^{4}C_{1}/^{7}C_{3} = 12/35
P(All 3 balls are green) = P(X = 3) = ^{3}C_{3}/^{7}C_{3} = 1/35
Hence, the probability distribution for this problem is given as follows
X 0 1 2 3 P(X) 4/35 18/35 12/35 1/35
Question 2: From a lot of 10 bulbs containing 3 defective ones, 4 bulbs are drawn at random. If X is a random variable that denotes the number of defective bulbs. Find the probability distribution of X.
Solution:
Since, X denotes the number of defective bulbs and there is a maximum of 3 defective bulbs, hence X can take values 0, 1, 2, and 3. Since 4 bulbs are drawn at random, the possible combination of drawing 4 bulbs is given by 10C4.
P(Getting No defective bulb) = P (X = 0) = ^{7}C_{4}/^{10}C_{4} = 1/6
P(Getting 1 Defective Bulb) = P (X = 1) = ^{3}C_{1} Ã— ^{7}C_{3}/^{10}C_{4} = 1/2
P(Getting 2 defective Bulb) = P(X = 2) = ^{3}C_{2} Ã— ^{7}C_{2}/^{10}C_{4} = 3/10
P(Getting 3 Defective Bulb) = P(X =3) = ^{3}C_{3} Ã— ^{7}C_{1}/^{10}C_{4} = 1/30
Hence Probability Distribution Table is given as follows
X 0 1 2 3 P(X) 1/6 1/2 3/10 1/30
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Conclusion – Probability Distribution
In conclusion, a probability distribution serves as a theoretical model of how likely various outcomes are to occur. While a frequency distribution deals with observed data, detailing the occurrences of different values within a sample, a probability distribution assigns probabilities to each possible outcome, reflecting the likelihood of encountering those values.
FAQs on Probability Distribution
What is Probability Distribution in statistics?
Probability Distribution is a function in statistics that shows how the probabilities for the random variable are distributed over a defined range.
What is Random Variable?
Random Variable is a real-valued function whose domain is sample space and the range is a real number. It is important for the knowledge of the possible number of events that occurred in the trial.Â
What is the Difference between Expectation and Variance?
Expectation is the mean of the independent outcomes of the random variable and is represented by Î¼ while variance is the measure of scatteredness of the random variable and is represented by Ïƒ.
What is the probability distribution formula?
It can be expressed as F(x) = the probability of X being less than or equal to x or (F(x) = P (X â‰¤ x))
What are the Types of Probability Distribution?
There are two types of Probability Distribution
- Discrete Probability Distribution – It takes finite values in the range.
- Continuous Probability Distribution – It takes continuous value in the range.
What are the Conditions for Probability distribution?
Any Probability distribution should satisfy two conditions:
- The probability of an event should be greater than 0.
- The sum of all the probabilities should be equal to 1.
What is the rule for probability distribution?
Step 1 involves assessing whether each probability falls within the range of 0 to 1, inclusive. Step 2 entails verifying if the total sum of all probabilities equals 1. Finally, Step 3 concludes that if both Step 1 and Step 2 conditions are met, the probability distribution is considered valid.
What is the probability distribution used for?
Probability Distribution has a number of uses and applications in real life such as business, engineering, medicine and other major sectors. It is mainly used to make future predictions based on a sample for a random event.
What is the importance of Probability distribution in Statistics?
It has a large number of applications in business, engineering, and other major sectors. It is mainly used to make future predictions based on a sample for a random experiment.
What is the importance of Probability distribution in Statistics?
The Poisson Probability Distribution is used for counting rare events that occur at a known average rate (Î») in a fixed interval. It’s helpful for modeling things like customer arrivals, disease outbreaks, or accident rates.