Question

what is the standard deviation of the random variable x "the number that comes up on the die"?

Answer 1

Assuming a fair 6-sided die, the random variable $X$ giving the number that comes up follows a uniform distribution with PMF

$P(X=x)=\begin{cases}\dfrac16&\text{for }x\in\{1,2,3,4,5,6\}\\\\0&\text{otherwise}\end{cases}$

The standard deviation of $X$ is the square root of the variance of $X$, $V[X]$. We have a formula for the variance in terms of the expected value, $E[X]$:

$V[X]=E[X^2]-E[X]^2$

where

$E[X]=\displaystyle\sum_xx\,P(X=x)=\sum_{x=1}^6\frac x6=\frac72$

$E[X^2]=\displaystyle\sum_xx^2\,P(X=x)=\sum_{x=1}^6\frac{x^2}6=\frac{91}6$

Then the variance is

$V[X]=\dfrac{91}6-\left(\dfrac72\right)^2=\dfrac{35}{12}$

so the standard deviation is

$\sqrt{V[X]}=\sqrt{\dfrac{35}{12}}\approx1.71$

Answer 2

Answer:2.0

Step-by-step explanation:

Just did it on a p e x