Question

Let random variable X ∼ U(0, 1). Let Y = a + bX, where a and b are constants.

(a) Find the distribution of Y .

(b) Find the mean and variance of Y .

(c) Find a and b so that Y ∼ U(−1, 1).

(d) Explain how to find a function (transformation), r(), so that W = r(X) has an exponential distribution with pdf f(w) = e^ −w, w > 0.

Answer #1

a) Suppose that X is a uniform continuous random variable where
0 < x < 5. Find the pdf f(x) and use it to find P(2 < x
< 3.5).
b) Suppose that Y has an exponential distribution with mean 20.
Find the pdf f(y) and use it to compute P(18 < Y < 23).
c) Let X be a beta random variable a = 2 and b = 3. Find P(0.25
< X < 0.50)

Let X be a random variable of the mixed type having the
distribution function
F ( x ) = 0 w h e r e x < 0
F ( x ) = x 2 4 w h e r e 0 ≤ x < 1
F ( x ) = x + 1 4 w h e r e 1 ≤ x < 2
Question 1: Find the mean of X
Question 2: Find the variance of X
Question...

Let X ∼ UNIF(0, 1). Find the pdf of Y = −5 ln(X) using the
transformation technique. Note that Y is an exponential random
variable. What is its parameter? Show your work.

Let
X and Y be random variable follow uniform U[0, 1]. Let Z = X to the
power of Y. What is the distribution of Z?

Suppose the random variable (X, Y ) has a joint pdf for the
form
?cxy 0≤x≤1,0≤y≤1 f(x,y) = .
0 elsewhere
(a) (5 pts) Find c so that f is a valid distribution.
(b) (6 pts) Find the marginal distribution, g(x) for X and the
marginal distribution for Y , h(y).
(c) (6 pts) Find P (X > Y ).
(d) (6 pts) Find the pdf of X +Y.
(e) (6 pts) Find P (Y < 1/2|X > 1/2).
(f)...

Let X be a random variable with pdf given by fX(x) = Cx2(1−x)1(0
< x < 1), where C > 0 and 1(·) is the indicator
function.
(a) Find the value of the constant C such that fX is a valid
pdf.
(b) Find P(1/2 ≤ X < 1).
(c) Find P(X ≤ 1/2).
(d) Find P(X = 1/2).
(e) Find P(1 ≤ X ≤ 2).
(f) Find EX.

1. Let (X,Y ) be a pair of random variables with joint pdf given
by f(x,y) = 1(0 < x < 1,0 < y < 1).
(a) Find P(X + Y ≤ 1).
(b) Find P(|X −Y|≤ 1/2).
(c) Find the joint cdf F(x,y) of (X,Y ) for all (x,y) ∈R×R.
(d) Find the marginal pdf fX of X. (e) Find the marginal pdf fY
of Y .
(f) Find the conditional pdf f(x|y) of X|Y = y for 0...

Let X be an exponential random variable. Suppose E[X|X>a]=b,
where b>a>0 are two constants. Compute the probability
P(X>a|X>a).

Let X be a random variable with pdf f(x)=12,
0<x<2.
a) Find the cdf F(x).
b) Find the mean of X.
c) Find the variance of X.
d) Find F (1.4).
e) Find P(12<X<1).
f) Find PX>3.

Let X and Y be continuous random variable with joint pdf
f(x,y) = y/144 if 0 < 4x < y < 12 and
0 otherwise
Find Cov (X,Y).

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