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Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a...
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...
(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...
(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...
Let A={1,2,3,4}. Pick a subset B⊆A uniformly among the 2^4 subsets (i.e. the power set ofA) and let X be its size. Then likewise pick a subset 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). Hint: 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).
3. Let X be a discrete random variable with the probability mass function 2 18 x=1,2,3,4, zero otherwise. , 12 a Find the probability distribution of Y-g(X- TXI b) Does Hy equal to g(Hx)? Ax=E(X), μ,-E(Y).
1. (6 pts) Consider a non-negative, discrete random variable X with codomain {0, 1, 2, 3, 4, 5, 6} and the following incomplete cumulative distribution function (c.d.f.): 0 0.1 1 0.2 2 ? 3 0.2 4 0.5 5 0.7 6 ? F(x) (a) Find the two missing values in the above table. (b) Let Y = (X2 + X)/2 be a new random variable defined in terms of X. Is Y a discrete or continuous random variable? Provide the probability...
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set of the following subsets of R (i) A = { n+1 n : n ∈ N} (ii) B = (0, 1] (iii) C = {x : x ∈ (0, 1), x is a rational number (b) Let X denote the indiscrete topology. Find the limit point set A 0 of any subset A of X. (c) Prove that a subset D of X is...
Let X and Y be two discrete random independent random variables. p(x) = 1/3 for x =-2,-1,0 p(y) = 1/2 for y =1,6 K = X + Y
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that X is geometric with parameter p = P(X = 1) if and only if the memoryless property P(X = n + m | X > n) = P(X = m) holds. To show that the memoryless property implies that X is geometric, you need to prove that the p.m.f. of X has to be P(X = k) = p(1 - p)^(k-1). For this, use...