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- The joint probability function of two discrete random variables X and Y is given by Ax,y) = c(2x+y), where x and y can assume all integers such that 0< xFor any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.
- Let X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.If Z is a discrete random variable with possible values of -1, 0, and 1, and the probability mass function is given by P(Z = -1) = 0.4, P(Z = 0) = 0.3, and P(Z = 1) = 0.3, what is the standard deviation of Z?Let X be a random variable with probability mass function P ( X = 1 ) = 1/2 , P ( X = 2 ) = 1/3 , a n d P ( X = 5 ) = 1/6 . Then E[1/x]=?
- (a) Let F denote the cumulative distribution function (cdf) of a uniformly distributed random variable X. If F(2) = 0.3, what is the probability that X is greater than 2 ? (b) Let F denote the cdf of a uniformly distributed random variable X. If F(2) = 0.3, and F(3) = 0.6, what is F(6) ? (c) Suppose X and Y are Poisson Random Variables. X has a mean of 1 and Y has a mean of 2. X and Y are correlated with CORR (X,Y)=0.5. whats the variance of X+YIf y is random variable has a probability mass function defined as follows find 1. and 2.3) The joint probability function of two discrete random variables X and Y is given by f(x; y) =c(2x + y), where x and y can assume all integers such that 0 ≤ x ≤ 2; 0≤ y ≤ 3, and f(x; y) = 0 otherwise. The constant value is 0.024Find P(X≥ 1, Y≤ 1). Give your answer to three decimal places
- Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1=(X1/r1)/(X2/r2) and Y2=X2. (a) Find the joint pdf of Y1 and Y2.Let U1, ....U5 be independent and standard uniform distibuted random variables given by P(U1 ≤ x) = x, 0 < x < 1 1. Compute the moment generating function E(e sU ) of the random variable U1. 2. Compute the moment generating function of the random variable Y = aU1 + U2 + U3 + U4 + U5 with a > 0 unknown. 3. Compute E(Y ) and V ar(Y ). 4. As an estimator for the unknow value θ = a we migth use as an estimator θb = 2 n Xn i=1 Yi − 4 = 2Y − 4. with Yi independent and identically distributed having the same cdf as the random variable Y discussed in part 2. Compute E(θb) and V ar(θb) and explain why this estimator is sometimes not very useful. 5.Give an upperbound on the probability P(| θb− a |> ) for every > 0.(Hint:Use Chebyshevs inequality!)Consider a random variable, Y , which has a quasi-Bernoulli structure. With probability p ∈ [0, 1] it takes value 0. With probability (1 − p) it is described by a continuous random variable, X , with the following PDF, f_X (x)=1+x, x∈[−1,0), −1≤x<0f_X (x)=1−x, x∈[0,1], 0≤x≤1f_X (x)=0, otherwise , x>1 Obtain CDF of Y, F_Y (y), and draw the sketch.