# E(x) of exponential random variable, joint density, and elevators +cov of multinomial

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• Apr 28th 2009, 08:28 PM
Cheezburger
E(x) of exponential random variable, joint density, and elevators +cov of multinomial
So, I've been working on a bonus review for a few hours now and have hit places where I'm not sure and places where I don't know where to start. I don't need help on all of these but it's simpler if I just post the whole pdf:
Beef Supreme - Now with cheese!

#3b I can't figure out... it's a classic problem that I've done before, only with n people, not n-separated-by-gender. If it was just n people, I'd do $\displaystyle 1 - \frac{(365)(364)...(364-n+1)}{365^n}$ but I don't know how to separate that population by gender.

#5 I have no idea where to start. I don't know how I'll end up with a factorial.

#6a I used a double integral, the outer between 0 and 1, the inner 0 and x, f(x,y)dydx... with first-time attempted LaTeX reproduction:
$\displaystyle \int_0^1 \int_0^X f(x,y) \, dydx$
and ended up with $\displaystyle \frac{15}{56}$
Are those the correct bounds?

#6b I figured I had to find the marginal density of y first, so
$\displaystyle \int_0^2 \frac{6}{14} (2x^2 + xy) \, dx$ gave me $\displaystyle \frac{16}{7} + \frac{6}{7} y$ for $\displaystyle 0<y<2$
Now I think I have to take a (double?) integral of this to find the expected value, but I don't know what formula or bounds to start with.

#7 I can't wrap my brain around at all. I was thinking along the lines of E(x) would simply be trials*probability $\displaystyle 10 * \frac{1}{5}$ but there's no way it could be that simple, and E(X) = 2 doesn't sound right.

#8 I should note there's a typo here: The cov(X1, X2) should be equal to -np1p2 and not -mp1p2. I've studied my 'Lecture 18' notes, and can't get any clarity out of it.