4. Suppose that the random variable X has density function Jo, and suppose that Y = e-X. fx (x) = Find E(Y), the expected value of Y. 4e-4r, x < 0 0≤x.
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- Suppose that X ~ and variance 1, i.e., (Y X = x) ~ N(2x, 1). Find the marginal pdf of Y. N(0,1). Then, under the condition X = x for some r E R, the distribution of Y is normal with mean 2x3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?Find the set AnU. U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {4, 8, 9, 10} Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. ANU= { (Use a comma to separate answers as needed.) O B. ANU is the empty set.
- 6. Assume that X is a continuous random variable with probability density function (pdf) 6. { (I+2) 27 if -2Let Y₁, Y₂, . density function Yn denote a random sample from a uniform distribution with " 0 ≤ y ≤0, 0, elsewhere. For testing Ho: 0 = 0o vs. Ha: 000, show that the a-level Likelihood Ratio Test (LRT) gives RR = {Y(n) > 00} U {Y(n) < 00α = } . f(y|0) =IF it was XG (23) Independent of Xen G (13) Find Miley D-Probability density Function of the Variable YaX+Xq 2 -the Mean variance of the variableSuppose X is a gamma random variable gamma(a, B), where a > 1 and 3 > 0: x >0 f(x) = { Ta-1e-=/8 otherwise What value of a maximizes the density function of X? Find the moment-generating function of X. Let Y be an independently and identically distributed gamma random variable as X, and let Z = X +Y. Find the moment-generating function of Z. What is the distribution of Z?2. If the random variable X has the probability density function f(x) = (1-x²), -1 < x < 1, 3 4 what is the variance of 15X + 2?34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).13. Suppose a continuous random variable x has the probability density Sk(1+ x), 0 1.4) using distribution function, determine the probabilities that (d) x is less than 2.2 (e) between 1.4 and 1.6 (f) Calculate mean and variance for the probability density function.We have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} If U = Y-X2 , what is the support for the random variable U? What would fu(u) and Fu(u) be? Say U = Y/X, what is the support for the random variable U? What would fu(u) and Fu(u) be?6. Suppose that the random variables X and Y have joint probability density function given by x+y, 0SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON