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Let X be an exponential random variable such that P(X<26) = P(X > 26). Calculate E[X|X...
Let X be an exponential random variable such that P(X < 27) = P(X > 27). Calculate E[X|X > 23].
Let X be a discrete random variable that follows a Poisson distribution with = 5. What is P(X< 4X > 2) ? Round your answer to at least 3 decimal places. Number
3. (10 points) Let X be a continuous random variable with CDF for x < -1 Fx(x) = { } (x3 +1) for -1<x<1 for x > 1 and let Y = X5 a. (4 points) Find the CDF of Y. b. (3 points) Find the PDF of Y. c. (3 points) Find E[Y]
Problem 4. Let X be a random variable with EIXI4 < oo. Define μ1 = EX and Alk-E(X-μ)k, k 2, 3, 4, and then 03 = 쓺 (skewness), a,--2 (kurtosis) 3/2 (1) Show that if P(X- > z) = P(X-円く-r) for every x > 0, then μ3-0, but not the other way around. (2) Compute as and a when X is Binomial with parameter p, exponential with mean1, uniform on [O, 1], standard normal, and double exponential (fx (x)-(1/2)e-M).
1. Let X be a random variable with pdf f(x )-, 0 < x < 2- a) Find the cdf F(x) b) Find the mean ofX.v c) Find the variance of X. d) Find F (1.75) e) Find PG < x < +' f) Find P(X> 1). g) Find the 40th percentile.*
3-) Let ocr<1 o w UUUUU is probability destiny function of X random variable. a- ) Find PlOCXCI) b.) Find Pix > 15) UUUUUU ca) Find € (x) and Var(x) d-) Find the distribution function
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
6. Let X be a normal random variable with mean u = 10. What is the standard deviation o if it is known that p (IX – 101 <>) =
X Y Z iid Suppose for random variable X, P(X > a) - exp( random variable Y, P(Y > y) exp(-0y) for y > 0, and for random variable , P(Z > z)--exp(-фа) for z > 0. (a) Obtain the moment generating functions of X, Y and Z. (b) Evaluate E(X2IX > 1) and show it is equal to a quadratic function of λ. (c) Calculate P(X > Y Z) if λ-1, θ--2 and φ--3. -λα) for x > 0,...