1. Let X1, , X„ be independent RVs which are U(-1, 1), and define n Xn = Ext. 1=1 Use Chebyshev's inequality to estimate P(IX„I > XXXXXXXXXXHow large must n be for the bound to be less than or equal...

1. Let X1, , X„ be independent RVs which are U(-1, 1), and define n Xn = Ext. 1=1 Use Chebyshev's inequality to estimate P(IX„I > 0.05). How large must n be for the bound to be less than or equal to 0.05? 2. Let X be a RV with density function
(r + 1)(1 — x)r, 0

where r > 0. For n = 1,2, ... , compute E[Xnj (i.e., the nth moment of X). 3. Let X have a gamma distribution with parameters p > 1 and a. Find ER I X]. Compare with I/E[Xj. 4. Conditionally, given X = x, let Y E U(0, x), and let X have density fx(x) = 1/x2 for x > 1. Find the density of Y. 5. Consider two_radioactive particles that decay independently at the same rate A. If X and Y denote their lifetimes, then X and Y are both Exp(1/ot). Calculate the probability that X > 2Y. 6. Let X and Y be independent Exp(a)-distributed RVs. Show that Z = X/(1 + Y) has density
h(z) — z + 1 + aczia z > 0. a(z + 1)2