Question relating to joint PDFs

Here are my questions:

1. Let $$X$$ ~ Unif$$(0, 1)$$, and $$0. Also, let $$\begin{cases} Y = 1 & \text{if 0

Similarly, let $$\begin{cases} Z = 1 & \text{if a

Find the variance-covariance matrix of Y, Z.

What I tried: Note that P($$0) = $$F_X(b)$$ - $$F_X(0)$$ = $$\frac{x}{b}$$.

Similarly, note P($$a) = $$F_X(1)$$ - $$F_X(0)$$ = $$\frac{x-a}{x-b}$$.

Then, $$E(Y)$$ = $$1*P($$0<X<b$$)$$ + $$0*[$$1$$-P($$0<X<b$$)]$$ = $$\frac{x}{b}$$. Similarly, E(Z) = $$\frac{x-a}{1-a}$$. Here, I get stuck (and I don't even know if it's right).

• There are two not-so-short questions here. Please ask (2) in a separate thread. – gunes Jun 21 '20 at 18:28
• Where specifically are you having a problem? I'm not really sure on this one. – wolfies Jun 21 '20 at 19:25
• @BoJack does the answer help you in any way? – gunes Jun 23 '20 at 15:47
• They seem correct. – gunes Jun 23 '20 at 19:57
• Sounds good. I've done so! – Bo Jack Jun 23 '20 at 20:14

First of all, $$P(0 not $$x/b$$, i.e. not a function of $$x$$. Similarly, $$P(a. These are also equal to $$E[Y],E[Z]$$ respectively. Also, we have $$E[Y^2]=E[Y],E[Z^2]=E[Z]$$ for binary RV case. For the joint moment, $$E[YZ]=P(Y=1\cap Z=1)=P(a. I think you can follow from here.