Please answer both. . Suppose that Y is a random variable with distribution function below. 1-e-v/2,...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
4.4.19 Random variableX has PDE fx(a)-1/4 -1s-33, 0 otherwise Define the random variable Y by Y = h(X)X2. (a) Find E[X and VarX (b) Find h(E[X]) and Eh(X) (c) Find ElY and Var[Y .4.6 The cumulative distribution func- tion of random variable V is 0 Fv(v)v5)/144-5<7, v> 7. (a) What are EV) and Var(V)? (b) What is EIV? 4.5.4 Y is an exponential random variable with variance Var(Y) 25. (a) What is the PDF of Y? (b) What is EY...
Suppose the random variable X has probability density function (pdf) - { -1 < x<1 otherwise C fx (x) C0 : where c is a constant. (a) Show that c = 1/7; (b) Graph fx (х); (c) Given that all of the moments exist, why are all the odd moments of X zero? (d) What is the median of the distribution of X? (e) Find E (X2) and hence var X; (f) Let X1, fx (x) What is the limiting...
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
Define the random variable Y = -2X. Determine the cumulative distribution function (CDF) of Y . Make sure to completely specify this function. Explain. Consider a random variable X with the following probability density function (PDF): s 2+2 if –2 < x < 2, fx(x) = { 0 otherwise. This random variable X is used in parts a, b, and c of this problem.
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
Questionl The random variable X and Y have the following joint probability mass function: 0.14 0.27 0.2 0.1 0.03 0.15 0.1 a) Determine the b) Find P(X-Y>2). c) Find PX s3|Y20) d) Determine E(XY) e) Determine E(X) and E(Y). f) Are X and Y independent? marginal pmf for X and Y. Question 2 Let X and Y be independent random variables with pdf 2-y 0sxS 2 f(x)- f(p)- 0, otherwise 0, otherwise a) b) Find E(XY). Find Var (2X +...
-l0 1- e-2x x MO 2) The distribution function for a random variable X is f(x) x <0 Find a) the density function 2 b) the probability that X 4 c) the probability that -3 <x 6inotion
4. [10 pts] Let X be a random variable with probability density function if 1 < a < 2, 2 f(a)a 0 otherwise. Find E(log X). Note: Throughout this course, log = loge.
(1) Suppose that X is a continuous random variable with probability density function 0<x< 1 f() = (3-X)/4 i<< <3 10 otherwise (a) Compute the mean and variance of X. (b) Compute P(X <3/2). (c) Find the first quartile (25th percentile) for the distribution.