Chapter 6.8, Problem 17E

To determine

To Calculate: The probability that the random variable in a normal distribution lies with two standard deviation of the mean.

Answer to Problem 17E

For any normal distribution, the probability that the random variable lies within two standard deviations of the mean is $0.9544$

Explanation of Solution

Given Information: The random variable is normally distributed and the interval for which probability is to be calculated is from $\text{μ}-2\text{σ}$ to $\text{μ}+2\text{σ}$ .

Concept Used: The required probability is same for any normal distribution, that is, the probability is the same for any pair of values of $\text{μ}$ , $\text{σ}$ .

Calculation: Since the probability is the same for any value of mean and standard deviation, replace $\text{μ}$ and $\text{σ}$ with any numbers of choice.

For example, put $\text{μ}=0$ (symmetric about the $y$ -axis) and $\text{σ}=1$ .

Now the probability is given by

  $P\left(\text{μ}-2\text{σ}\le X\le \text{μ}+2\text{σ}\right)=P\left(-2\le X\le 2\right)=\underset{-2}{\overset{2}{{\displaystyle \int }}}\frac{1}{1\cdot \sqrt{2\text{π}}}{e}^{-\frac{x^2}{2}}dx$

Since the above integral contains $e^{-\frac{x^2}{2}}$ , which has no known anti-derivative, use a calculator or computer to get the probability as

  $P\left(-2\le X\le 2\right)=0.9544=95.44\%$

Therefore, for any normal distribution, the probability that the random variable lies within two standard deviations of the mean is $0.9544$