2. Joint pdf of random variables X and Y is given as 1 Pxy(x, y) = |— ^ cxp[= {(x|+|y|)] for «

3. The length of time required by students to complete a 1 hour exam is a random variable with a pdf given by: f (x) = = ca + for 0 sæ<1 a. Find c. Enter c as a reduced fraction. C = b. Find F(x). Enter coefficients as reduced fractions and use ^ to denote powers. F(x) = c. Find the probability that a student takes less than 30 minutes to complete the exam. Enter the probability as a reduced fraction. prob = d. Find the median length of time to take the exam. Enter your answer in hours with 4 decimal places. median = hours e. Find the length of time for the first 10% of the students to complete the exam. Enter your answer in hours with 4 decimal places. hours answer = f. Find the expected value, variance, and standard deviation of X. Enter your answer in hours with 4 decimal places (or hours squared in the case of variance), hours expected value =

Let X be a random variable with pdff(x) = 4x^3 if 0 < x < 1 and zero otherwise. Use thecumulative (CDF) technique to determine the pdf of each of the following random variables: 1) Y=X^4, 2) W=e^(-x) 3) Z=1-e^(-x) 4) U=X(1-X)

Suppose that f(x)=k/(4^x), x=1,2,3,... is the probability function for a random variable X. Find the value of 1/k.

let k be a random variable that takes with equal prob 1/(2n+1), values in integer the interral' [-n, n]. Find the PMF of the random variable y = log (x) where X = a¹" and a>o.

Suppose that X is a random variable for which the moment generating function is as follows: MX(t) = e^(2t^2+3t) for −∞ < t < ∞. Find the mean and variance of X. (b) Suppose that X has moment generating function MX(t) =(3e^t/4 + 1/4)^6(i) Find the p.m.f. of X. (ii) Find the mean and variance of X. (c) A person with some finite number of keys wants to open a door. He tries the keys one-by-one independently at random with replacement. How many trails you expect, from him, to open the door? (d) Obtain the form of moment generating function (m.g.f.) for the following p.m.f. – p(x) = ((2^x)(e^-2))/×!, x = 0,1,2, … . Also calculate the mean and variance from m.g.f.

the random variable Y is such that : E(2Y+3)=6    and    Var(2-3Y)=11 find      E(Y^2)

The moment generating function of the random variable X is given by mX(s) = e2e^(t)−2 and the moment generating function of the random variable Y is mY (s) =(3/4et +1/4)10. If it is assumed that the random variables X and Y are independent, findthe following:(a) E(XY)(b) E[(X − Y )2](c) Var(2X − 3Y)

Suppose that f (x) = e=* for 0 < x Determine the mean and variance of the random variable.

Suppose that f(x)=2/(3^x), x=1,2,3,... is the probability function for a random variable X. Find P(X>3). Use 4 decimal places.