This is all the information given in the question.
This is all the information given in the question. 2. For any Random Variable Y, show...
Problem 4 Let Yı, Y2, ..., Y, denote a random sample from the probability density function (0 + 1)ye f(0) = 0 <y <1,0 > -1 elsewhere Find the MLE for .
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)
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).
5. Y is a continuous random variable with pdf f(y) = (4 – y)/8, 0<y< 4. (a) Find E(Y). (b) Find E(Y2). (c) Find Var(Y).
[9] Given any two real numbers x and y such that x < y, show that there exists a rational number q such that x < a <y.
7.695 points Save Answer QUESTION 4 Let the random variable X and Y have the joint p.d.f. for 0 < x < 1, 0 < y < 1, and 0 < x +y < 1 | 24cy f(x, y) = { lo otherwise Find E[X].
2. (10 pts The random variables X and Y have joint density function f(x, y) == 22 + y2 <1. Compute the joint density function of R= x2 + y2 and = tan-1(Y/X).
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
6. (5 marks) Suppose Y is a Normal random variable. If P(Y <1)=0.25 and P(Y > 0)=0.9. what are y, and o?
Let Yı, Y, have the joint density S 2, 0 < y2 <yi <1 f(y1, y2) = 0, elsewhere. Use the method of transformation to derive the joint density function for U1 = Y/Y2,U2 = Y2, and then derive the marginal density of U1.