Question

continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY and W = X/Y. Are U and W independent random variables? P. 7 Consider 35 independent random variables, each following Geometric distribution with parameter 0.2. (Remember that the variance of a Geometric distribution with parameter p is (1-P)/p.) (a) State the Central Limit Theorem for the average of the random variables. (b) What is the probability that the average of the 35 random variables is between 4.5 and 5.2? (c) What is the probability that the average is less than 5.1 if it is greater than 4.7? (d) What is the threshold that the average exceeds with a probability of exactly 3%? (e) Extra: 3 points) Instead of 35, at least how many variables do we need if we want the average to exceed 5.5 with a probability of at most 3%? P.8 Extra: (7 points) The pdf of the x2 distribution of order r is fr(x) = "/2-4e-/2, > 0. Prov it by induction that it is the pdf of r number of squares of independent standard normal distribution In particular, assume so for r and show that the sum of a random variable with pdf fr(2) and anoth independent random variable with pdf fi(r) has the pdf fr+1(3). 9 Extra: (7 points) Derive the pdf of the Gamma distribution for integer orders k. Start from a Pois process and first obtain the cdf in terms of the pmf of a Poisson distribution
P7

Answer 1

Pod a centreal Limit Thorem Let X, X ., o le a random sample of size in - that in a sequence of independent and identically distrilented seandem variables drawn from a dis becibution of expected value giron ley fr. and finite variance given ley T? Let, Sn = x + x 2 + ... + xn . Then as n approaches infinity the scandom variables In (Sn - i) converge in distribution, to a Normal (0,02). du (S. M) (0 ) 6). . In (Sn - 1/0:2) dy N(o,). n 1-0:2 LV20 nehrer, Sr. 2 x P[ 4.5 < So <5. 27 - PI 135 (4.5-5) 135 (Sn - 5 135 (5:2-5)] 20 120 P[-066 < < < 0:26] 2 Plz < 0:26] P[2 <-0.66]. P[2 20:26] - I + P [24 0.66] - 0.602 5681 - 1 + 0 74 53731 20.34 79 412.

& P[ So < 51 | 5n34.7] • P[474 So <5-1] P[sa > 4.7]. P[su <5.1] - [5 < 47] P[ Sa > 4. 1] РІ pl also find 0) < 15 (5) ->). (1976-) 64157-8) 23 , (Sn-5) Sn - 5) 20 235 (s. V. 20 20 1- & Pr 35 (Sn - 5 20 135 (417-5) 20 - P[2 < 0.13] P[24 - 0:40] 1-P 24-0.40 - P[2<0.13] - 1 + P[24 0.40] PZ4 0.40] = 0.5517169 + 0.655 4211 - 0.6554212 2 0.31604

1 d. Let the there threshold la x: P[ 8.>x] = 0.03. 3) in p[ Sn< ] =bewczec0c94.0-03 2) - P[2< 153(].00 9P[<< 55(2-0). 0:47. ro ) 135 (2-5). 1.99. 120 x = 5+ 120 x 1.89 35 26.43. " The threeshold is 6.43. the sample size Let lan. & P[ So > 5.5 & 0.03. 5) PET Cons) > (95-9] <0-03 3) P[ z > Nã xos 1 30.03 3) P[z< where >.97 - P[z <1.89). -) Os }, -89 m pg? x 20 • 285.763 ~ 286. 20 0.52 " Atleast 286 variables neill lee meedee.