Question

From the probability distribution, find the standard deviation for the random variable x, which represents the number of cars per household in a town of 1000 households.
x P(x)
0 0.125
1 0.428
2 0.256
3 0.108
4 0.083

Answer 1

Compute the first and second moments. The first moment is the same as the expected value or mean. The second moment is involved in computing the variance.

First moment:

$E[X]=\displaystyle\sum_xx\,P(x)=0\cdot0.125+1\cdot0.428+\cdots+4\cdot0.083$

$E[X]=1.596$

Second moment:

$E[X^2]=\displaystyle\sum_xx^2\,P(x)=0^2\cdot0.125+1^2\cdot0.428+\cdots+4^2\cdot0.083$

$E[X^2]=3.752$

The variance of $X$ is

$V[X]=E[(X-E[X])^2]=E[X^2-2X\,E[X]+E[X]^2]=E[X^2]-E[X]^2$

$V[X]=2.156$

The standard deviation is the square root of the variance:

$\sqrt{V[X]}\approx\boxed{1.468}$