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777dan777 [17]
2 years ago
15

Let

Mathematics
1 answer:
Digiron [165]2 years ago
5 0

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) = \sqrt{E(X^2) - V(X)} = \sqrt{23}

Expectation is linear, so

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

so that

V(Y) = V(2X+10) = (424 + 40\sqrt{23}) - (2\sqrt{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

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