# Sum of Normal Distributions

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A normal distribution is a probability distribution which can be clearly represented using a bell curve. A standard normal distribution is also a normal distribution, however the mean of such a distribution is 0 and the standard deviation is 1. A bell curve, also known as the normal distribution curve is a graph which depends on factors such as the mean and the standard deviation of set of real numbers. The z-score value of normal distribution is given by a formula which depends on the mean standard deviation and the normal random ‘x’ value.

Example 1: For a set of random variables, given the normal random variable to be equal to 15. If the mean of the given set is 5 and the standard deviation is 2, then what is the standard score or the z-score?

Given, x = normal random variable = 15

Mean, μ = 5

Standard deviation, σ = 2

The z-score = (x – μ)/ σ

This implies, z-score = (15 – 5)/ 2 = 10/2 = 5.

Therefore, the standard z-score is 5.

Example 2: For a set of random variables, given the normal random variable to be equal to 24. If the mean of the given set is 18 and the standard deviation is 3, then what is the standard score or the z-score?

Given, x = normal random variable = 24

Mean, μ = 18

Standard deviation, σ = 3

The z-score = (x – μ)/ σ

This implies, z-score = (24 – 18)/ 3 = 6/3 = 2.

Therefore, the standard z-score is 2.