If X is a random variable with characteristic function ox (w) and if the 'r'th moment exists, it can be determined by a* [¢x (@)] m, = (- j)" () = 0
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- let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x isProve that if M(t) is the MGF of a random variable X, then the MGF of a + bX is e^at M (bt)3. Let X and Y be continuous random variables with joint PDF (3x 0s ysxs1 f(x, y) = {* otherwise Determine the correlation of variables X and Y.
- 3. The length of time required by students to complete a 1 hour exam is a random variable with a pdf given by: f (x) = = ca + for 0 sæ<1 a. Find c. Enter c as a reduced fraction. C = b. Find F(x). Enter coefficients as reduced fractions and use ^ to denote powers. F(x) = c. Find the probability that a student takes less than 30 minutes to complete the exam. Enter the probability as a reduced fraction. prob = d. Find the median length of time to take the exam. Enter your answer in hours with 4 decimal places. median = hours e. Find the length of time for the first 10% of the students to complete the exam. Enter your answer in hours with 4 decimal places. hours answer = f. Find the expected value, variance, and standard deviation of X. Enter your answer in hours with 4 decimal places (or hours squared in the case of variance), hours expected value =2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.Derive the MGF for an exponential random variable X ~ exp(X) (try the Kernel trick!) t-入 ele' –1) A- t
- Suppose the random variable Y has a mean of 21 and a variance of 36. Let Z = √36 Show that #z=0. Show that o₂ = 1. (Y-21). #₂ = E(Y-D] -0--0 (Round your responses to two decimal places) o = var (Y-] )- (Round your responses to two decimal places)let k be a random variable that takes with equal prob 1/(2n+1), values in integer the interral' [-n, n]. Find the PMF of the random variable y = log (x) where X = a¹" and a>o.Let X be a random variable with the function, -(x -A) "for x > 2, Let f (x) = { elsewhere. Derive u and show that T = X is biased estimator of 2 . Can you modify T =X in order to get an unbiased estimator of 2.
- Suppose that X is a random variable for which the moment generating function is as follows: MX(t) = e^(2t^2+3t) for −∞ < t < ∞. Find the mean and variance of X. (b) Suppose that X has moment generating function MX(t) =(3e^t/4 + 1/4)^6(i) Find the p.m.f. of X. (ii) Find the mean and variance of X. (c) A person with some finite number of keys wants to open a door. He tries the keys one-by-one independently at random with replacement. How many trails you expect, from him, to open the door? (d) Obtain the form of moment generating function (m.g.f.) for the following p.m.f. – p(x) = ((2^x)(e^-2))/×!, x = 0,1,2, … . Also calculate the mean and variance from m.g.f.Suppose X is a random variable with the cdf Fy(x) = 1-(1+x²) x>0, c>0, Y>0 Derive the pdf of the inverse of X, Y=1/x²The moment generating function of the random variable X is given by mX(s) = e2e^(t)−2 and the moment generating function of the random variable Y is mY (s) =(3/4et +1/4)10. If it is assumed that the random variables X and Y are independent, findthe following:(a) E(XY)(b) E[(X − Y )2](c) Var(2X − 3Y)