STAT2011 Sample Questions Semester 1 2020
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STAT2011 Sample Questions
Semester 1 2020
Shorter questions
1. (a) (P) Find A n B n C if A = {x : 0 s x s 6}, B = {x : 4 s x} and C = {x = 0, 1, 2, 3, . . .}.
(b) (P) Ten basketball players meet in the school gym for a pickup game. How
many ways can they form a blue team and a red team of five players each?
2. (a) (C) A desk has three drawers. The first contains two gold coins, the second has two silver coins, and the third has one gold coin and one silver coin. A coin is drawn at random from a drawer selected at random. Suppose the coin selected was silver. What is the probability that the other coin in that drawer is gold?
(b) (C) A bleary-eyed student awakens one morning, late for an 8:00 class, and
pulls two socks out of a drawer that contains two black, four brown, and two blue socks, all randomly arranged. What is the probability that the two he draws are a matched pair?
3. (P) Let X have a Poisson distribution with a mean of 5. Find P (2 s X < 4).
4. (P) It is known that the probability is 0.3 that a child exposed to a certain contagious disease will catch it. What is the probability that the 10th child exposed to the disease will be the fourth to catch? Assume that children are independent.
5. (C) Let Y1 , Y2 , . . . , Y9 be a random sample of size 9 from a continuous distribution with pdf fY (y) = 2y, 0 s y s 1. Given that its cdf is FY (y) = y2 , 0 s y s 1, find the pdf of Y3(′) , the third smallest in the sam- ple.
6. (C) Let fXY (x, y) be the joint pdf of two continuos random variables given by fXY (x, y) = 2, x + y < 1, x > 0; y > 0. Find fY |x(y), the conditional probability density of Y given X.
Longer questions
7. A total of n different letters, say addressed to person P1 , P2 , . . . , Pn are assigned at random to n labeled envelopes, say with address A1 , A2 , . . . , An .
(a) (P) If n = 3, characterise all sample outcomes in the sample space S3 .
(Hint: write (2,1,3) for the sample outcome that A1 contains letter to P2 , A2 contains letter to P1 and A3 letter to P3 .)
(b) (P) For any n > 2, show that the number of sample outcomes in the sample
space is Sn = n!.
(c) (C) Let the random variable Xi e {0, 1} and Xi = 1 when envelope with
address Ai contains letter to Pi and Xi = 0 otherwise. Show that
n一1
pX乞 (k) = P(Xi = k) = . 1 n
, n
for k = 0
for k = 1 .
(d) (D) Derive the joint distribution of (Xi , Xj ) for i j. (Hint: P (Xi = 1, Xj = 1) = (n1一1)n .)
(e) Let X = X1 + X2 + . . . + Xn , calculate (i) (P) E(X), (ii) (C) E(X1X2 ) and
(iii) (C) Var(X1 + X2 ).
8. (D) Let X 1 and X2 be the means of two independent samples of sizes n1 and n2 from a normal population with the mean µ and variance σ 2 . Show that U = ωX1 + (1 - ω)X2 is also unbiased for µ, where 0 < ω < 1 is a constant.
Find the value of ω such that Var(U) is minimum.
2022-06-01