Problem #3: Let Z be the standard normal random variable.Problem #3(a):Problem #3(b):Problem #3(c):(a) Find the value of the probability density function fx(x) of X = |Z + 1 for x = 1.(b) Find the value of the probability density function fy(v) of Y = e² for y = 2.(c) Find E[Y] (Hint: this problem is closely related to the moment generating function for the normaldistribution.)Enter your answer symbolically,as in these examplesEnter your answer symbolically,as in these examplesEnter your answer symbolically,as in these examples

Given that Z~N(0,1). The PDF of Z is fZ(z)=12πe-z22 for -∞<z<∞. The CDF of Z is FZ(z)=P(Z≤z)…

Answered: Problem #3: Let Z be the standard…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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(a) Suppose (X,Y) is a pair of continuous random variables with joint probability density function if s > 0 and t>1 fx,y(s, t) = { 32 (b) Let Fx,y be the joint cumulative distribution function of the pair (X, Y). Compute Fx,y(4, 5). if not Let (X, Y) be as in Part (a). The random variable log(Y) has an exponential distribution. What is the corresponding rate parameter?
Let Z be the standard normal random variable. (a) Find the value of the probability density function fx (x) of X = [Z + 11 for x = 1. (b) Find the value of the probability density function fy(y) of Y = e² for y = 2. (c) Find E[Y] (Hint: this problem is closely related to the moment generating function for the normal distribution.)
2. A random variable X has a probability density function (pdf) given by , 0 |X > ).
Probability distribution.
(a) The discrete random variable X has moment generating function: Mx (1) = k + = (1² + P²³ + P²³), where k is a positive constant. (1) Find the value of k. (ii) Determine P (X > 4). (iii) Use Mx (1) to calculate the mean and the variance of X. (b) The continuous random variable X has probability density function -1 = {kx (² kx (x-3), 3< x < 4 0, otherwise. (1) Find the value of k. (ii) Find the mean of X. (iii) Find the variance of X. f(x) =
1. Let f (x) = ce I(0,2] (x)+ce¹-¹1(2₁00) (x) where c is some constant. (a) Find the constant c that will make f a probability density function. (b) Find the corresponding cumulative distribution function F. (c) Find the expected value of X.
Example 20 : If the probability density function of X is given by x/ 2, for 0 < x<1 1/2, for 1< x <2 f (x) ={ (3 – x) / 2, for 2
Let X and Y have the following joint distribution:X \Y 0 1 2 0 2/12 1/12 5/12 1 1/12 1/12 2/12(a) Find the probability distribution of X|Y = 0, and E[X|Y = 0].(b) Find E[Y ] and E[X].(c) Find the probability that Y is larger than zero.(d) Are X and Y dependent? Justify your answer
(9) If X is a continuous random variable with p. d. f. (a) 60 (b) 1 + f(x) {P The probability density function of Y = 5X + 3 is: 2y-1 300 12 ,0 ≤ x ≤ 3 , otherwise (c) y²-y-6 200 lor (d) 1/1 + 12/1/202
4 (b) An insurance company provides customers with both auto and home insurance policies. For a particular customer, Χ is the deduction on his or her auto policy and Y is the deduction on the home policy. Possible values of Χ are K100 and K250, and for Y are K0, K100 and K200. The joint probability density function for ( ) ,YΧ is given by the following table: Χ Y K100 K250 K0 0.20 0.05 K100 0.10 0.15 K200 0.20 0.30 iv. If we look only at those insurance customers selecting the lowest auto mobile insurance deduction (K100), what is the probability that a randomly selected\ customer will also select the lowest home deduction (K0). v. Compute the correlation coefficient of Χ and Y
Q1 (a) Given probability distribution function for discrete random variable X as below: 2 4 8 10 P(X = x) 0.08 0.25 0.35 0.2 0.12 (i) Find the mean of X. (ii) Find the variance of X. (b) Let the continuous random variable X has probability density function as follows, (k(x + 2)², -2

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