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.

Answer #1

Let X ∼ UNIF(0, 1). Find the pdf of Y = 1 − X using the
distribution-function technique. Also indicate the support of
Y.

Let X have a uniform distribution on (0, 1) and let y = -ln ( x
)
a. Construct the CDF of Y graphically
b. Find the CDF of Y using CDF method
c. Find the PDF of Y using PDF method

Let ? be a random variable with a PDF
?(?)= 1/(x+1) for ? ∈ (0, ? − 1). Answer the following
questions
(a) Find the CDF
(b) Show that a random variable ? = ln(? + 1) has uniform ?(0,1)
distribution. Hint: calculate the CDF of ?

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.

For X1, ..., Xn iid Unif(0, 1):
a) Show the conditional pdf X(i)|X(j) ∼ X(j)Beta(i, j − i)
b Let n=5, find the joint pdf between X(2) and X(4).

Let X and Y have the pdf f(x, y) = 1, 0 < x < 1, 0
< y < 1, zero elsewhere.
Find the cdf and pdf of the product Z =
X+Y.

Let X ∼ Geo(?) with Θ = [0,1].
a) Show that pdf of the random variable X is in the
one-parameter
regular exponential family of distributions.
b) If X1, ... , Xn is a sample of iid Geo(?) random variables
with
Θ = (0, 1), determine a complete minimal sufficient statistic
for ?.

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).

Let X and Y be random variables with the joint pdf
fX,Y(x,y) = 6x, 0 ≤ y ≤ 1−x, 0 ≤ x ≤1.
1. Are X and Y independent? Explain with a picture.
2. Find the marginal pdf fX(x).
3. Find P( Y < 1/8 | X = 1/2 )

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and
0<=y<=1.
a) Find k.
b) Find the joint cumulative density function of (X,Y)
c) Find the marginal pdf of X and Y.
d) Find Pr[Y<X2] and Pr[X+Y>0.5]

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