Suppose that Y, and Y, are uniformly distributed over the triangle shaded in the accompanying figure. y2 (0, 1) (-1, 0) (1, 0) (a) Find (b) Find P(Y, - Y2 2 0).
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- Let Y1 and Y2 be independent and uniformly distributed over the interval (0,1). Find P(4Y1<Y2)LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Suppose X1,…,Xn,…are identically distributed with mean E(X1)=μ<∞ and Var(X1)=σ2<∞. In addition, we assume that Cov(Xk,Xk+1)=0 for k=1,2,… but Cov(Xk,Xj)=0 whenever ∣k−j∣≥2. (a) Find the limiting distribution ofXˉn=n−1i=1∑nXiasn→∞. (b) Find the limiting distribution of Zn=∑nnXin+∑n=1nXiX,eias n→∞. (c) LetY1,Y2… be i.i.d random variables with mean 0 and variance 1 . Additionally, letXk=Yk+Yk+1 for ,k≥1. Find the limiting distribution of Xˉn⋅
- Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0If X1, X2, ... , Xk have the multinomial distribution ofDefinition 8, show that the covariance of Xi and Xj is−nθiθj for i = 1, 2, ... , k, j = 1, 2, ... , k, and i Z j.
- Suppose X has uniform distribution on (−1, 1) and, given X = x, Y is uniformly distributed on (−(1-x^2)^1/2, (1+x^2)^1/2). Find the join denisty of X and Y.Suppose {X1, X2, …, X7, Y} are independent RVs. Each Xj is Rayleigh with mean and Y = P(2) is Posson. Let (X1, …, X7) iid and find E(X) 2. Var(X) 3. E(X-2Y) 4. Var(X-2y)Let X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.
- The distribution of a random vector is given by a function:f(x,y) = y/2 - x/2 ; for 0 < x < 1 and 2 < y < 3 = 0 ; otherwise Determine P( X > Y - 2), and sketch the graf.a) Find the marginal pmfs of X and Y b) Find the conditional pmf of X given Y = 1Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.