Question

Let

X
be a random variable such that
E
(
X
2
)
=
81
and
V
(
X
)
=
58
. Compute V
(
2
X
+
10
)
E
(
2
X
+
10
)

Answer 1

What I gather from the question is that
`X` has second moment
`E(X^2)=81` and variance
`V(X) = 58`, and you're asked to find the expectation and variance of the random variable
`Y=2X+10`.

From the given second moment and variance, we find the expectation of
`X` :

`V(X) = E(X^2) - E(X)^2 \implies E(X) = √(E(X^2) - V(X)) = √(23)`

Expectation is linear, so

`E(Y) = E(2X+10) = 2 E(X) + 10 = \boxed{2√(23) + 10}`

Using the same variance identity, we have

`V(Y) = V(2X+10) = E((2X+10)^2) - E(2X+10)^2`

and

`E((2X+10)^2) = E(4X^2 + 40X + 100) = 4E(X^2) + 40E(X) + 100 = 424 + 40√(23)`

so that

`V(Y) = V(2X+10) = (424 + 40√(23)) - (2√(23) + 10)^2 = \boxed{232}`

Alternatively, we can use the identity

`V(aX+b) = a^2 V(X) \implies V(2X+10) = 4V(X) = 232`