Look at #7 5. Let X be exponential with Care page 6 - 0.25 and a....
4. Let X and Y be independent exponential random variables with pa- rameter ? 1. Given that X and Y are independent, their joint pdf is given by the product of the individual pdfs of X and Y, that is, fxy(x,y) = fx(x)fy(y) The joint pdf is defined over the same set of r-values and y-values that the individual pdfs were defined for. Using this information, calculate P(X - Y < t) where you can assume t is a positive...
7. Let h(T) =T,IT <1 (and equal to zero otherwise) be an impulse response function of a WSS process. Find its transfer function (v).
Exercise 4.9. Let X ~ Poisson(10). (a) Find P(X>7). (b) Find P(X < 13 X > 7).
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x > 0) and f(y) = µe-µy (y > 0) respectively. (i) Let Z = min(X, Y ). Find f(z), E(Z), and Var(Z). (ii) Let W = max(X, Y ). Find f(w) (it is not an exponential pdf). (iii) Find E(W) (there are two methods - one does not require further integration). (iv) Find Cov(Z,W). (v) Find Var(W).
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
7. Let x(t) be a Poisson process having rate 6 5. a) P(X(1)=2] b) P(X(2) = 31x(1)s 2] c) P(X(2) = 31x(4) = 5] d) EIX(1)] e) Var[X(1)] (1D 7. Let x(t) be a Poisson process having rate 6 5. a) P(X(1)=2] b) P(X(2) = 31x(1)s 2] c) P(X(2) = 31x(4) = 5] d) EIX(1)] e) Var[X(1)] (1D
5. If X and Y are independent and identically distributed with Exponential(A), compute El and 6. Let R be the region bounded by the points (0, 1), (-1,0) and (1,0). Joint pdf of (x, Y) is: 1, if (r,y) e R 0, otherwise. Compute P(X-1, γ 7. If X U(0,1) and Y U(0, 1) independent random variables, find the joint pdf of (X+y,x -Y). Also compute marginal pdf of X+Y 8. If x Ezpomential(0.5) and Y ~ Erponential0.5) independent random...
Let X be uniform on [0, 1], and let Y be exponential with rate λ, so that P(Y ≥ t) = e ^(−λt ) t ≥ 0 and 1 if t < 0 Assume that X and Y are independent, and define W = X + 2Y . a) For any w ≥ 0 and x ∈ [0, 1], compute P(W ≥ w|X = x) b) By undoing the conditioning on X, use the result from part (a) to compute...
exp(1) 7. (15 points) Consider two independent, exponential random variables X,Y Let U = X + Y and V = X/(X+Y). (a) (5 points) Calculate the joint pdf of U and V. (b) (5 points) Identify the distribution of U. If it has a "named" distribution, you must state it. Otherwise support and pdf is enough. (c) (5 points) Identify the distribution of V.If it has a “named” distribution, you must state it. Otherwise support and pdf is enough.
Let X be a Poisson (mean = 5) and Let Y be a Poisson (mean = 4). Let Z = X + Y. Find P( X = 3 | Z = 6). Assume X and Y are independent. Show answers for P(A), P(B), P(AB), and and hence P(A|B). Here A = [X = 3], B = [Z = 6]