(7) Suppose that X₁ and X₂ are two random variables with j.m.g. f. 2 (t₁ - 1)(t₂-1) + (t₂ − 1)² ' MX₁X₂ (t) then var (X₁) is 1/2. От O F =
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- X1 and X2 are independent random variables such that Xi has PDF fXi(x)={λiexp(−λix) when x≥0, 0 otherwise}. What is P[X2<X1]?Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?
- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )
- 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) = 0A continuous random variable X is defined by: Solve: 1. f(x) = (3+x)²/16 ; -3≤ x≤ -1 2. f(x) = (6 - 2x)²/16 ; -1≤x≤1 3. f(x) = (3 - x²)/16 ; -1≤x≤3If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- Suppose Xis a random variable with E(X) = 2, and E(X2) = 5. Compute the following quantities: a) Var(X) b) E(X−1)2 c) Var(−X−3) d) E(5X+ 4) PARTS A,B,C ARE ANSWERED ALREADY, JUST D PLEASEFind the variance by calculating the first two moments of the random variable X = (- 1 / λ) ln (1-U), where U ~ U (0,1) and λ> 0.2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.