Let X and Y be two jointly continuous random variables with...

Let X and Y be two jointly continuous random variables with joint pdf $\begin{equation*} f(x, y) = \begin{cases} x + y & 0 < x \le 1, \quad 0 < y \le 1\\ 0 & \text{else} \end{cases} \end{equation*}$ Find E[XY$^2$]

Transcript

00:01 Hi, i'm david and i'm going to help you answer your question.

00:04 Now let me bring up your question here.

00:06 In this question we're going to find the covariance and the correlation coefficient.

00:12 Let me remind you that the covariance of the xy, it will equal to the e of the xy minus e of the x times e of the y.

00:26 Also, the correlation coefficient of the x, y, it will equal to the covariance on the x, y, dividing by the sa variance on the x and then france on the y.

00:44 And here we will have to find the e of the xy first.

00:52 So e on the xy, it will equal to the integral, double integral here where we have the two inside.

01:03 Now we will try to compute the dx first, d -y after.

01:09 We see x goes from the...

01:12 Here we can write this down to the x smaller equal to 1 minus y.

01:17 So x goes from the 0 to the 1 minus y.

01:20 Y goes from the 0 up to infinity.

01:25 And then if we compute it, we get equal to 0.

01:27 To infinity and in some hand the true x evaluate from zero to the one minus y d y the again equal to from zo to infinity here we have the two minus two y and then d y therefore we have this one will be the two y minus the y square evaluating from zero to infinity and then we will be the two we will have this will equal to the...

02:07 Let me see it's not infinity, it should equal to one only because it cannot equal to infinity here.

02:13 Sorry about that.

02:15 So we have one here and there's an also one as well and one.

02:23 So if we put the one inside we will have the 2 minus 1 and equal to 1.

02:30 So that will be the e of the xy.

02:34 Now to find the e .m...

Question

Let X and Y be two jointly continuous random variables with joint pdf $\begin{equation} f(x, y) = \begin{cases} x + y & 0 < x \le 1, \quad 0 < y \le 1\\ 0 & \text{else} \end{cases} \end{equation}$ Find E[XY$^2$]

Question

Let X and Y be two jointly continuous random variables with joint pdf $\begin{equation*} f(x, y) = \begin{cases} x + y & 0 < x \le 1, \quad 0 < y \le 1\\ 0 & \text{else} \end{cases} \end{equation*}$ Find E[XY$^2$]

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Let X and Y be two jointly continuous random variables with joint pdf $\begin{equation} f(x, y) = \begin{cases} x + y & 0 < x \le 1, \quad 0 < y \le 1\\ 0 & \text{else} \end{cases} \end{equation}$ Find E[XY$^2$]