finding expected value using integration

[forest badge]

Can someone explain the solution to part c

I don't get how to solve it by integration

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[umbral fog]

That’s the standard for finding expectation of continuous RVs

Then just applying linearity of expectation for this specific question

[forest badge]

The only thing I know is the E(X)=∫xf(x)

Can you please tell me how to get to the answer step by step

[forest badge]

What's that

[umbral fog]

Splitting up the expectation

Into 2E(H^2) + 3E(G) + 3

First term was worked out from using the data given on variance and the second term is the one calculated by integrating

[forest badge]

Oh we can do that?

I thought we could only do something like E(3x)=3E(x)

[forest badge]

Thank you so much

[umbral fog]

Yeah but also E(X+Y) = E(X) + E(Y) so you can split up expressions

That’s linearity

Just be aware E(XY) doesn’t equal E(X)E(Y) unless X and Y are independent

Which is why the E(H^2) part can’t be reduced to simply E(H)^2

[forest badge]

Got it👍