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- Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with density curve f(x) = (1/3)e(-x/3) for x > 0 and f(x) = 0 otherwise. What is the probability that response time exceeds 5 seconds? What is the probability that response time exceeds 10 seconds?Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln square root(X)] using Jensen’s inequality.
- If X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.If X has a standard normal distribution (X ∼ N(0, 1)), then the characteristic function of X is ϕX(t) = e −0.5(t^2) . Using this fact, show that X has skewness 0 and kurtosis 3
- suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)Let X1, X2,...Xn be a random sample of size n from a normal distribution with mean u and variance o2. Let Xn denote the sample average, defined in the usual way. PROVE E [Xn] = uWhat is the Marginal PDF of f(x)=2? Is the event independent?
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