Answer
Mean: $22.5$
Variance: $206.25$
Standard deviation: $14.36$
Work Step by Step
Random variable $X$ has a discrete uniform distribution with the parameters:
$$
a=0, b=9
$$
Calculate the mean and the variance:
$$
\mathbb{E}(X)=\frac{0+9}{2}=4.5
$$
$$ \operatorname{Var}(X) =\frac{(9-0+1)^{2}-1}{12}=8.25 $$
$$ \sigma_{X} =\sqrt{\operatorname{Var}(X)}=2.87 $$
Now calculate the mean, variance and standard deviation for the random
variable $5 X :$
$$
\mathbb{E}(5 X) =5 \times \mathbb{E}(X)=[22.5] $$
$$ \operatorname{Var}(5 X) =\mathbb{E}\left((5 X-\mathbb{E}(5 X))^{2}\right)=\mathbb{E}\left((5 X-5 \mathbb{E}(X))^{2}\right) $$
$$ =5^{2} \times \mathbb{E}\left((X-\mathbb{E}(X))^{2}\right)=25 \times \operatorname{Var}(X)=206.25 $$
$$\sigma_{5 X}=\sqrt{\operatorname{Var}(5 X)}=\left[\begin{array}{l}{14.36}\end{array}\right]$$
