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Question 3 14 pts The cdf of random variable Wis Fw(w) = { 1 + 20...
The probability density function for random variable Wis given as follows: 120 w>0 20 Let x be the 100pth percentile of W and y be the 100(1-p)th percentile of W, where o<p〈1. Express y as a function of x. ln(1-e- x/20 ) 20 -x/20 In 1-e 20 Cy- -20 ln 1-ex20 y20 n e-/20
Question 6 A random variable X has cdf χ20 Plotthe cdf and identif.,(x)-1-0.2~ a) Plot the cdf and identify the type of the random variable. b) Find the pdf of X. c) Calculate P[-4eX<-1], P(xS2], P(X=1], Pf2-K6], and P[X>10]. d) Calculate the mean and the variance of X. If the random variable X passes through a system with the following chara cteristic function: e) f) Find the pdf of Y. Calculate the mean and the variance of Y. Good Luck
1. Consider the following Cumulative Distribution Function (CDF) of random variable 0.41 1 t <3 0.78 3 < t < 5 0.94 5t<7 F(t) = a. 4 Find P(T> 3); P(1.5 < T b. [3] Find E(3T +5) and V (3T5) 6); P(T < 5IT2)
20 pts total] Consider the random variable X with the following CDF shown below a. [04 pts] Determine the correct value for c. b. [04 pts] Find P[1 s X s 3]. c. (04 pts] Find E[X], the expected value of x d. (04 pts] Find Var[X], the variance of X. e. [04 pts] Find the second moment of X.
(1 point) Given the MGF of random variable Wis M(t) = €2.4+-+15t provide the name of the distribution of W, as well as its parameters. Select all that apply. There may be more than one correct answer. A. normal(u = 2.4, o2 = 15) B. normallu = 15, o2 = 2.4) C. gamma(a = 4.8,B = 15) D. gamma(a = 15, B = 4.8) E. normal(u = 4.8, 02 = 15) OF. gamma(a = 15, B = 2.4) G. normallu...
The distribution function of a random variable X is given by 0 Fw={ F(2) = 1+2 <-1 -1<r<1 => 1 "iszai (a) (5 points) Find the p.d.f(f(x)) of X (b) (5 points) Find P(0.3 < X <0.5)
Question 3 [10 marks Let W Then the p.d.f. 1 fw (w) 2"/21 (n/2) exp(-w/2) w3-1, w>0. and the c.d.f. is denoted as Fw (w) (a) Show that 0, n > 0, and (i) The function fw(w) is a p.d.f. (i.e., that fw(w) 2 0 for w Jo fw(w)dw 1). (ii) The mode of W is n - 2 for n > 2. (b) As n oo, W becomes normally distributed with mean n and variance 2n. This has led...
Let X be a random variable with CDF z<0 G()=/2 0 <IS2 z>2 1 Suppose Y = X2 is another random variable, find (a) P(1/2 X 3/2), (b) P(1s X< 2) (c) P(Y X) (d) P(X 2Y). (f) If Z VX, find the CDF of Z. (d) P(X+Y 3/4)
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).
The random variable X has CDF 0 x<-1, 0.2 -1s<O, 0.7 OS<1, 1 21. Fx () (a) Draw a graph of the CDF. (b) Write Px(x), the PMF of X. Be sure to write the value of all a from -oo to oo.