TOPIC:Mean,variance and cdf of random variables.
2. For a continuous density f(x) find u, oand the cumulative distribution function F(x) Ch 4/3, 6, 24 f(x)= a. eux,...
(15 points) Let X be a continuous random variable with cumulative distribution function F(x) = 0, r <α Inr, a< x <b 1, b< (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
3) The continuous random variable X has the probability density function, ), 2 3x3 f(x) = { a, 35x55 2 - bx, 5 < x < 6 elsewere 10 i)Find the value of a and b and hence, sketch f(x) ii) Find the cumulative distribution function, f(x) and sketch it.
6. Let X be a continuous random variable whose probability density function is: 0, x <0, x20.5 Find the median un the mode. 7. Let X be a continuous random variable whose cumulative distribution function is: F(x) = 0.1x, ja 0S$s10, Find 1) the densitv function of random variable U-12-X. 0, ja x<0, I, ja x>10.
Let X be a continuous random variable with cumulative
distribution function F(x) = 1 − X−α x ≥ 1
where α > 0. Find the mean, variance and the rth moment of
X.
Question 1: Let X be a continuous random variable with cumulative distribution function where a >0. Find the mean, variance and the rth moment of X
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F() = 0, <a Inx, a < x <b 1, b<a (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
be a continuous random variable with probability density function 3. Let for 0 r 1 a, for 2 < < 4 0, elsew here 2 7 fx(x) = (a) Find a to make fx(x) an acceptable probability density function. (b) Determine the (cumulative) distribution function F(x) and draw its graph.
4. (30pts) A continuous random variable X has the probability density function: hx - 1 sx 32 f(x) =Jo-hx 2 x 3 0 x >3 which ean bo graphed as f(x) 1 2 a) Find h which makes f(x) a valid probability density function b) Find the expected value E(X) of the probability density function f(x) c) Find the cumulative distribution function F(x). Show all you work
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)
Question 3: Let X be a continuous random variable with
cumulative distribution function FX (x) = P (X ≤ x). Let Y = FX
(x). Find the probability density function and the cumulative
distribution function of Y .
Question 3: Let X be a continuous random variable with cumulative distribution function FX(x) = P(X-x). Let Y = FX (x). Find the probability density function and the cumulative distribution function of Y