Question

Let X be a random variable with density function f(x) = 1/4 for -3 <= x <= 5, and 0 otherwise. Find the density of Y = X^2 and of Y = (X - 1)^2, of Y = |X-1|, and of Y=(X-1)^4.

Answer #1

Let X be a random variable with density function f(x) = 2 5 x
for x ∈ [2, 3] and f(x) = 0, otherwise. (a) (6 pts) Compute E[(X −
2)3 ] without attempting to find the density function of Y = (X −
2)3 . (b) (6 pts) Find the density function of Y = (X − 2)3

Let X be a random variable with probability density function
f(x) = {3/10x(3-x) if 0<=x<=2
.........{0 otherwise
a) Find the standard deviation of X to four decimal
places.
b) Find the mean of X to four decimal places.
c) Let y=x2 find the probability density function
fy of Y.

Let the probability density function of the random variable X be
f(x) = { e ^2x if x ≤ 0 ;1 /x ^2 if x ≥ 2 ; 0 otherwise}
Find the cumulative distribution function (cdf) of X.

Let X be the random variable with probability density function
f(x) = 0.5x for 0 ≤ x ≤ 2 and zero otherwise. Find the
mean and standard deviation of the random variable X.

The density function of random variable X is given by f(x) = 1/4
, if 0
Find P(x>2)
Find the expected value of X, E(X).
Find variance of X, Var(X).
Let F(X) be cumulative distribution function of X. Find
F(3/2)

Part A
The variable X(random variable) has a density function of the
following
f(x) = {5e-5x if 0<= x < infinity and 0
otherwise}
Calculate E(ex)
Part B
Let X be a continuous random variable with probability density
function
f (x) = {6/x2 if 2<x<3 and 0 otherwise }
Find E (ln (X)).
.

2. Let the probability density function (pdf) of random variable
X be given by:
f(x) = C (2x -
x²),
for
0< x < 2,
f(x) = 0,
otherwise
Find the value of
C.
(5points)
Find cumulative probability function
F(x)
(5points)
Find P (0 < X < 1), P (1< X < 2), P (2 < X
<3)
(3points)
Find the mean, : , and variance,
F².
(6points)

Let X be a random variable with probability density function
f(x) = { λe^(−λx) 0 ≤ x < ∞
0 otherwise } for some λ > 0.
a. Compute the cumulative distribution function F(x), where F(x)
= Prob(X < x) viewed as a function of x.
b. The α-percentile of a random variable is the number mα such
that F(mα) = α, where α ∈ (0, 1). Compute the α-percentile of the
random variable X. The value of mα will...

Let X and Y be two continuous random variables with joint
probability density function
f(x,y) =
6x 0<y<1, 0<x<y,
0 otherwise.
a) Find the marginal density of Y .
b) Are X and Y independent?
c) Find the conditional density of X given Y = 1 /2

Let X be a random variable with density f X ( x ) = ( 1 / 2 )
cos x for x ∈ [ − π / 2 , π / 2 ]. (a) Show that this is a valid
density function. (b) What is the distribution function of Y = sin
X? (c) What is the density function of Y?

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