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Probability Distribution Function

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Probability Distribution Function is a function in mathematics that gives the probability of all the possible outcomes of any event. We define probability distribution as how all the possible probabilities of any event are allocated over the distinct values for an unexpected variable.

Two functions are used to describe the Probability Distribution of the function which includes, Probability Mass Function and Probability Distribution Function. Here in this article, we will learn about Probability Distribution Function. There are two ways to represent the data, Discrete Data and Continuous Data, and based on that we can represent the Probability Distribution Function into two further categories.

In this article, we will learn about Probability Distribution, Probability Distribution Function, its Formula, Graphs, related Examples, and others in detail.

What is Probability Distribution?

Probability Distribution is defined as the function that gives the probability of all the possible values of the random variables. There are various types of probability distribution, which are Binomial Distribution, Bernoulli Distribution, Normal Distribution, and Geometric Distribution.

Probability Distribution Definition

Probability Distribution is basically the set of all possible outcomes of any random experiment or event . It basically tells how the probability of an event is distributed for different values of random variables.

Probability Distribution Function

Probability Distribution Function is defined as the function that is used to express the distribution of a probability. It is also called Cumulative Distribution Function(CDF). For any random variable X where its value is evaluated at the points ‘x’, then probability distribution function gives the probability that X takes the value less than equal to x. We represent the probability distribution as, F(x) = P (X ≤ x)

Different types of probability, they are expressed differently. These functions are also used for Probability Density Functions for different variables.

Cumulative Probability Distribution for closed interval (a, b] is given as 

P(a < X ≤ b) = F(b) – F(a)

Note: For probability distribution function the value of the variable lies between 0 and 1.

Probability Distribution Function Formula

Probability distribution function formula gives the probability of the all the possible outcomes of any random variable. On the basis of types of random variable the different formula for the probability distribution function are,

Probability Distribution of a Discrete Random Variable

Discreate Random Varibale are the variable that takes distinct countable values are 0, 1, 2, 3 … The formula for probability distribution of a discrete random variable is,

Probability Distribution Function:

F(x) = P (X ≤ x)

Probability Distribution of a Continuous Random Variable

Continuous Random Variable is a variable that takes the infinitely many values. The formula for probability distribution of a continuous random variable is,

Probability Distribution Function: F(x) = P (X ≤ x)

Probability Density Function:

f(x) = d/dx (F(x))

where,

  • F(x) = ∫-∞x f(u)du

Normal Probability Distribution Formula

It is also understood as Gaussian diffusion and it directs to the equation or graph which are bell-shaped. The formula for a standard probability distribution is as expressed:

P(x) = (1/√(2)πσ²)e−(x − μ)²/2σ²

where,

  • μ is the Mean
  • σ is the Standard Distribution
  • x is the Normal random variable

Note: If mean(μ) = 0 and standard deviation(σ) = 1, then this distribution is described to be normal distribution.

Binomial Probability Distribution Formula

It is defined as the probability that occurred when the event consists of “n” repeated trials and the outcome of each trial may or may not occur. The formula for binomial probability is as stated below:

P(r out of n) = n!/r!(n − r)! × pr(1 − p)n – r = nCr × pr(1 − p)n−r

where,

  • n is the Total number of events
  • r is the Total number of successful events
  • p is the Probability of success on a single trial
  • 1 – p is the Probability of failure

Note: nCr =  n!/r!(n – r)!

Probability Distribution Graph

The graph that plot the Probability Distribution Functions are called the Probability Distribution graphs. These graphs helps us to visualise the probability distribution around a random variable and help us to easily find the required solution.

The sum of all the probabilities in any discrete distribution is one and the for continuous distribution of random variable the area under the graph is equal to 1. The distribution graph of continuous distribution function is added below, where X (the random variable) lies between a and b. It is made using Probability Density Function

Probability-Density-Function

For discreate random variables the probability distribution is given using Bernoulli distribution.

Probability Distribution Function and Probability Density Function

We easily describe the Probability Ditribution using Probability Distribution Function and Probability Density Function. Using a probability distribution function is very useful for both continuous probability distribution and discrete probability distribution, while probability density function(pdf) is only used for continuous probability distribution.

Read More,

Probability Distribution Function Example

Example 1: Suppose we toss two dice. Make a table of the probabilities for the sum of the dice. The possibilities are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Solution:

Probability Distribution Table

X P(x)
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

Example 2: The number of old people living in houses on a randomly selected city block is described by the following probability distribution.

Number of adults Probability
           (x)     P(x)
3 0.50
4 0.25
5 0.10
6 ?

What is the probability that 6 or more old peoples live in a randomly selected house?

Solution:

Sum of all the p(probability) is equal to 1

Probability that six or more old peoples live in a house,

= 1 – (0.50 + 0.25 + 0.10)

= 0.15

Thus, probability that six or more old peoples live in a house is equal to 0.15

Example 3: When a fair coin is tossed 8 times, Probability of:

  • Exactly Four Heads
  • At least Four Heads

Solution:

Every coin tossed can be considered as the Bernoulli trial. Suppose X be the number of heads in this experiment,

n = 8

p = 1/2

So,

P(X = x) = nCx pn – x (1 – p)x, x =  0, 1, 2, 3,…n

P(X = x) = 8Cxp8 – x(1 – p)x

P(Exactly 4 Heads)

= P(x = 4)

= 8C4 p4 (1 – p)4

= 8!/4!4!(1/2)4(1/2)4

= (8 × 7 × 6 × 5/2 × 3 × 4) × (1/16) × (1/16)

= 420/1536

= 35/128

Thus, the probability of Exactly Four Heads in a Eight Coin Toss experiment is 35/128

P(At Least 4 Heads)

= P(X >= 4)

= P(X = 4) + P(X = 5) + P(X = 6)+ P(X = 7) + P(X = 8).

= 8C4 p4 (1 – p)4 + 8C5 p3 (1 – p)5 + 8C6 p2 (1 – p)6 + 8C7 p1(1 – p)7 + 8C8(1 – p)8

= 8!/4!4!(1/2)8 + 8!/5!3!(1/2)8 + 8!/6!2!(1/2)8 + 8!/7!1!(1/2)8 + 8!/8!(1/2)8

= 8 × 7 × 6 × 5/4 × 3 × 2 × 256 + 8 × 7 × 6/3 × 2 × 256 + 8/256 + 1/256

= 1680/6144 + 336/1536 + 9/256

= 70/256 +  56/256 + 9/256

= 135/256

Thus, the probability of Atleast Four Heads in a Eight Coin Toss experiment is 135/256

Example 4: Calculate the probability of getting 10 heads, if a coin is tossed 12 times.

Solution:

Given,

  • Number of Trials(n) = 12
  • Number of Success(r) = 10 (getting 10 heads)
  • Probability of Single Head(p) = 1/2 = 0.5

To find nCr =  n!/r!(n – r)!

=  12!/10!(12 – 10)!

=  (12 × 11 × 10!)/10!2!

= 66

To find pr = (0.5)10 = 0.00097665625

So, the probability of getting 10 heads is:

P(x) = nCr pr (1 – p)n – r

= 66 × 0.00097665625 × (1 – 0.5)(12-10)

= 0.0644593125 × (0.5)2

= 0.016114828125

The probability of getting 10 heads = 0.0161…

Example 5: Suppose that each time you take a free throw shot, you have a 35% chance of making it. If you take 25 shots, what is the probability of making exactly 15 of them?

Solution:

Given,

  • n = 25
  • r = 15
  • p = 0.35
  • q = 0.65

Compute

C25,15 (0.35)15 (0.65)10 = 0.165

There is a 16.5% chance of making exactly 15 shots.

Example 6: There is a total of 5 people in the room, what is the possibility that someone in the room shares His / Her birthday with at least someone else?

Solution:

P(s) = p(At least someone shares with someone else)                        

P(d) = p(No one share their birthday everyone has a different birthday)

p(s) + p(d) = 1 or 100%

p(s) =100% – p(d)

There are 5 people in the room, the possibility that no one shares his/her birthday

= (365 × 364 × 363 × 362 × 361) ⁄ (365)5

= (365! ⁄ (365 – 5)!) ⁄ 3655

= (365! ⁄ 360!) ⁄ 3655

= 0.9728

p(d) = 0,9728 or 97.28%

p(s) = 100% – p(d)

= 100% – 97.28% or 1 – 0.9728

= 2.72%  ≈ 0.0272

Practice Questions on Probability Distribution Function

Q1: Find the Probability Distribution of Number of Heads when two coins are tossed Simultaneously.

Q2: What is the Probability Distribution of number of Kings when three cards are drawn at random.

Q3: A die is thrown twice. Find the probability getting number of sixes.

Q4: A coin is thrown until a tail appears or head appears three times continuously. Find the probability distribution of tosses.

FAQs on Probability Distribution Function

1. What is Probability Distribution Function?

A function that relates all the possible outcomes of an experiment with their corresponding probabilities is called the Probability Distribution Function.

2. Why Do We Need a Probability Distribution Function?

The probability distribution function is the integral of the probability density function. This function is very useful because it tells us about the probability of an event that will occur in a given interval.

3. What Are Main Properties of Probability Distribution?

Probability Distribution Function has Various properties and some of the properties of the Probability Distribution Function are,

  • Probability of each function is between zero and one.
  • The sum of the probabilities of all the related functions is one.

4. What are Main Types of Probability Distributions?

There are various types of probability distribution and some of the common types of probability distribution include,

  • Binomial Distribution
  • Poisson Distribution
  • Normal Distribution
  • Uniform Distribution, etc.

5. What is Difference between Probability Mass Function (PMF) and Probability Density Function (PDF)?

The basic difference between Probability Mass Functions (PMF) and Probability Distribution Functions (PDF) is PMFs are used to describe discrete probability distributions whereas PDFs are used to describe continuous probability distributions.



Last Updated : 01 Mar, 2024
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