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- (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+YLet X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- If X has the Poisson distribution with P(X=1) = 2P(X=2), then P(X ≥ 2) is approximately:Consider a process consisting of a linear trend with an additive noise termconsisting of independent random variables wt with zero means and variancesσ2w, that is,xt = β0 + β1t + wt,where β0, β1 are fixed constants.(a) Prove xt is nonstationary.(b) Prove that the first difference series ∇xt = xt − xt−1 is stationary byfinding its mean and autocovariance function.(c) Repeat part (b) if wt is replaced by a general stationary process, say yt,with mean function µy and autocovariance function γy(h)If X is Poisson random variable with parameter λ, compute E[1/(X + 1)]
- Find P(X=4) if X has a Poisson distribution such that 3P(X=1)=P(X=2)X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and 1/2, respectively. What is the moment generator function M(t) of X?Suppose that X is a discrete random variable with P(X = 0) = 0.3, P(X = 1) = 0.2, P(X =2) = 0.15, and P(X = 3) = 0.35. Graph the frequency function and the cumulative distribution function of X.
- (a) Suppose rain is falling at an average rate of 30 drops per square inch per minute. What is the chance that a particular square inch is not hit by any drops during a given 10-second period? What assumptions are you making? (b) Let X and Y be independent Poisson random variables with parameters 1 and2 respectively. Find (i)P(X=1andY =2) (ii)P(X+Y/2)>=1Let W1, W2, . . . be an uncorrelated random sequence with mean 0 and variance 1.Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn(n ∈ N) with a and X0 given. For each of the following two separate cases, find the mean functionmX (n) (n ∈ N) and covariance function CX (m, n) (m, n ∈ N) for the process {Xn : n ∈ N}, anddetermine if it is wide-sense stationary.a. a = 1 and X0 = 0.b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated withW1, W2, . . . .6 - If the random variable X has a Poisson distribution with parameter 9, which of the following is the probability of P(X<14) using the central limit theorem? a) 0.6826 B) 0.8413 NS) 0.9332 D) 0.1826 TO) 0.0336