6-x-4, 0x<2 0 1 2cych Exri If for two R.V. s X&Y the joint pdf is given by, otherwise Find Frix (o (1), Frix (alt), Ely/x-1]. var [Ylx-i] = E[^\x-]- (E[1\x=1])!
> + x 0. x)e-0, f(x) = fall + x)e-0x tu function. That is, shou is, show that > where (a) Show that f(x) is a density fun What is f(x) > 0, and that bo f(x) dr = 1. (b) Find ELI (c) Find Var(37)
8. Let 02 (1)e, 00; x > 0 -0x f(x) = 1 +0 (a) Show that f(x) is a probability density function (b) Find P(X > x) (c) Find the failure rate function of X
17. If F is a d.f. such that F(0-) = 0, then -F)) dx xdF(x) +oo Thus if X is a positive r.., then we have P[X x}dx E(X) PX> x}dx =
Let X have a gamma distribution with parameters a > 2 and 3 > 0. (a) Prove that the mean of 1/X is B/(a - 1). (b) Prove that the variance of 1/X is 82/[(a - 1)(a 2)].
If X has a gamma distribution with alfa = 2 and beta = 1, find pl 1.6 < X <2.8) Goma Distidiosa Fox) = f(...) for x 30 Q2. Let X be a continuous random variable with f(x) = 0.5ež for x > 0 and 0 otherwise. Find the expected value of g(x) = ei x <2.8) for xxo Gumma. Distribution F() =fQ, XB) - Ba otherwise
3. An infinite bar has initial temperature distribution: T(x,0)-T0 [s(x-l) +δ(x + 1)] Find T(x,t) for >0. The free space Green's function for the 1-D diffusion equation is G(x -x)- e 4DI 4TDt
1. Let X1,.. X, be a random sample from -0x re 0= 01, where Find a most powerful test of size a for Ho 01>>0. 00 against H 0
8. The Gamma(a, A) distribution has density f(x)(a) where for a0' a 0 and A > 0 (a) Showfx,of(x) dr-1. Recall !"(a)-C"rtta-idt. (b) If X has a gamma distribution with parameters α and λ, find a general expression for E(Xk). (Answer: ) (c) Use your answer to the last question to find Var(X). The identity「(α + 1) a「(a) will help.
O A Gamma random variable x with parameters xson Bo has the following moment generating function: M(t) = (1 - 45 for taß (9 pts) . Use the reg.f. Mit given above E(X4), (where X is a Canan c.v.) to calculate (9 pts) [ Use the mg.f. Mit given above to calculate the standard deviation of a Gamma rov. X. (9 pts) © The skewness of a r.V. X, with mear=u& std. dev. = T, is given by : skewness...