Probability: Correlation

Hi, I have tried to do this question but I get stuck at the last part where it is asked of me to find the correlation. Can anyone help me? Thank you for your time

Reply 1

ghostwalker

17

Original post by gardash

Hi, I have tried to do this question but I get stuck at the last part where it is asked of me to find the correlation. Can anyone help me? Thank you for your time

In what way are you stuck? It seems to be a case of just applying a formula

Reply 2

zeratul

1

\text{Corr}(Y_1,Y_2)=\dfrac{ \text {Cov}(Y_1,Y_2)}{\sqrt{\text{Var}(Y_1)\text{Var}(Y_2)}}=\dfrac{ \mathbb {E} [Y_1Y_2]- \mathbb {E}[Y_1]\mathbb{E}[Y_2]}{\sqrt{\text{Var}(Y_1)\text{Var}(Y_2)}}

coming either from definitions or from immediate applications of definitions.

The expectations and variances of the Y_i come from their marginal distributions, and E[Y_1Y_2] comes from their joint distribution.

Hope this is along the lines of what you expected to see.

(edited 10 years ago)

Reply 3

gardash

OP

Original post by zeratul

\text{Corr}(Y_1,Y_2)=\dfrac{ \text {Cov}(Y_1,Y_2)}{\sqrt{\text{Var}(Y_1)\text{Var}(Y_2)}}=\dfrac{ \mathbb {E} [Y_1Y_2]- \mathbb {E}[Y_1]\mathbb{E}[Y_2]}{\sqrt{\text{Var}(Y_1)\text{Var}(Y_2)}}

coming either from definitions or from immediate applications of definitions.

The expectations and variances of the Y_i come from their marginal distributions, and E[Y_1Y_2] comes from their joint distribution.

Hope this is along the lines of what you expected to see.

I was stuck on the part E[Y_1Y_2]. How do I find it?

Reply 4

zeratul

1

In complete generality, if f(x,y) is the joint density of the random variables in the region D of the x,y plane, and h(x,y) is just an integrable function, then