Question

Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C C B uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X - Y). Note: X,Y can take value 0 if you pick the empty set. You can either write down a table or a compact expression of the form P(X = i, Y = j). (b) (2 pts) Verify that for j > 1: р n-j +1 P(Bin(n, p) = j) х * P(Bin(n,p) = j - 1) 1-p Now use this fact to determine the value of j for which P(Bin(n,p) = j) is highest.

Answer 1

Solution:

Given data:

a)

= {P{X P[X = 1, Y = y) = P[X = x]P[Y = y|X = ] x = 0, 1, 2, 3, 4; y = 0, 1,... I 0 otherwise (1) () 1 = 0, 1, 2, 3, 4; y = 0, 1,... I 0 otherwise 24 24 2

Clearly,
X ~ Bin(4,
and
Y|X = r ~ Bin(x,

E(X –Y)= Ex Ey X(X –Y) = Ex(X – Eyjx(Y))= Ex(X – X ExG) 4x 1

b)

Please give me thumbup....

n! (n – j)!}}P" (1 – p)"; pu "(1 – p)-j+1 P[Bin(n,p) = j] = p n! (1 - p) (n - j)!!" р (n-1 +1) n! (1-P) (n - 1 + 1)!(1 - 1)! p (n-1 +1) -P[Bin(n,p) = (1 - 1)] (1-P) j '(1 – p}"-j+1 =