   Chapter 8.5, Problem 19E

Chapter
Section
Textbook Problem

For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.

To determine

To find: The probability that the random variable lies within two standard deviations of the mean.

Explanation

Given information:

The function is normal distribution.

Calculation:

Show the probability density function for normal distribution as follows.

f(x)=1σ2πe(xμ)22σ2 (1)

Consider the standard means for normal distributions are μ2σ and μ+2σ .

Find the probability density that the random variable lies within two standard deviations of the mean P(μ2σXμ+2σ) as shown below.

P(μ2σXμ+2σ)=μ2σμ+2σ1σ2πe(xμ)22σ2dx (2)

Consider t=xμσ (3)

Differentiate both sides of the Equation.

dt=1σdx

Calculate the lower limit of t using Equation (3).

Substitute μ2σ for x in Equation (3)

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