(6) Let X and Y be random variables with var(X) = 4, var(Y) = 9 and %3D var(X – Y) = 16. Then cov(X,Y) is -. F
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- 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-Y2X1 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]?Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?
- 2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.Show that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .A 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).?
- If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j. (a) What is Cov(Xi,Yi) in terms of σx, σy and ρ? (b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi (c) Determine Cov(X ̄,Y ̄). B2. Consider the random variables Xi and Yi from question B1 again. (a) Show that the sample covariance is an unbiased estimator of Cov(X1,Y1). Hint: consider the equality Xi − X ̄ = (Xi − μ) − (X ̄ − μ). (b) Can you conclude from the statement in part (a) that the sample correlation is an unbiased estimator of Corr(X1, Y1)? Justify your answer.Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Find 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.Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variables