What is the expected value and variance of a random variable that is based on another random variable?

Question

Im unsure if this is the correct question, sorry.

Let B be the a random variable with expected value 10 and variance 4 that is defined on B >= 0.

If Y = 4 + B/10, what is the expected value and variance of Y? Y is defined on 0 - 24.

Thanks.

Answer 1

Expectation is linear. This means that

$$E[aX + bY] = aE[X] + bE[Y]$$

for random variables $X$ and $Y$ and $a, b \in \mathbb{R}$

We have a similar property for variance, which is true when $X$ and $Y$ are independent.

$$Var[aX + bY] = a^2Var[X] + b^2Var[Y]$$

We also have that $E[a] = a$ and $Var[a] = 0$ (the expectation of a constant is just the constant, and the variance of a constant is zero).

So in your case we have $$E[Y] = E[4 + \frac{1}{10}B] = E[4] + \frac{1}{10}E[B] = 4 + \frac{10}{10} = 5$$

and

$$Var[Y] = Var[4 + \frac{1}{10}B] = Var[4] + \frac{1}{100} Var[B] = 0+ \frac{4}{100} = 0.04$$

Answer 2

We can use the linearity of expectation to find $E(Y) = E(4+ B/10) = E(4) + E(B/10) = 4 + \frac{1}{10}E(B) = 4 + 10/10 = 5.$

Similarly for the variance, we have $Var(Y) = Var(4 + B/10) = \frac{1}{100}Var(B) = 4/100 = 1/25.$