How to Find Probability Using the Mean and Variance in Normal Distribution?

You can calculate the probability in a normal distribution using the z-score formula: P(X<x)=Φ(x–μ​)/σ, where Φ is the cumulative distribution function, x is the value, μ is the mean, and σ is the standard deviation.

To calculate the probability in a normal distribution given the mean (μ) and variance (σ²), you can use the z-score formula along with the standard normal distribution. The formula is:

P (X<x) = Φ (x–μ​)/σ

Here’s a detailed explanation of the steps involved:

1. Understand the Components:
   • P(X < x): This represents the probability that a random variable X in a normal distribution is less than a specific value x.
   • Φ: This symbolizes the cumulative distribution function (CDF) of the standard normal distribution.
   • (x–μ​)/σ​: This part calculates the z-score, representing how many standard deviations a particular value x is from the mean μ in terms of the standard deviation σ.
2. Calculate the Z-Score:
   • Subtract the mean (μ) from the specific value x.
   • Divide the result by the standard deviation (σ).
   • The z-score ((x–μ​)/σ) tells you how many standard deviations the specific value x is from the mean.
3. Use the Standard Normal Distribution Table:
   • Once you have the z-score, you can use a standard normal distribution table to find the cumulative probability.
   • The cumulative probability (Φ) gives the probability that a standard normal random variable is less than or equal to the calculated z-score.
4. Interpret the Result:
   • The final result P(X < x) is the probability that a random variable X in the normal distribution is less than the specific value x.

Example: Suppose you have a normal distribution with a mean (μ) of 50 and a variance (σ²) of 25. You want to find the probability that X is less than 55.

Calculate the z-score: (x–μ​)/σ = (55-50)/√25 = 5/5 = 1

Use the standard normal distribution table or calculator to find Φ (1). For [Tex]z = 1[/Tex], Φ (1) is approximately 0.8413.

Interpret the result: The probability P(X < 55) is approximately 0.8413, meaning there’s an 84.13% chance that a randomly selected value from the distribution is less than 55.