Question

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)

Answer #1

Suppose X is a continuous uniform random variable between −1 and
1, i.e., X ∼ U(−1, 1). Find the CDF and the PDF of P = −ln|X|.

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.

Let X represent a continuous random variable with a Uniform
distribution over the interval from 0 to 2. Find the following
probabilities (use 2 decimal places for all answers): (a) P(X ≤
1.92) = (b) P(X < 1.92) = (c) P(0.22 ≤ X ≤ 1.56) = (d) P(X <
0.22 or X > 1.56) =

If X is a discrete random variable with uniform distribution,
where f(x) > 0 when x = -1, 0, and 1(f(x) = 0, otherwise). If Y
is another discrete random variable with identical distribution as
X. In addition, X and Y are independent.
1. Please find the probability distribution of (X + Y)/2 and
plot it.
2. Please find variance of (X + Y)/2

Question 3 Suppose the random variable X has the uniform
distribution, fX(x) = 1, 0 < x < 1. Suppose the random
variable Y is related to X via Y = (-ln(1 - X))^1/3.
(a) Demonstrate that the pdf of Y is fY (y) = 3y^2 e^-y^3,
y>0. (Hint: Work out FY (y))
(b) Determine E[Y ]. (Hint: Use Wolfram Alpha to undertake the
integration.)

Suppose X is a random variable with pdf
f(x)= {c(1-x) 0<x<1
{0 otherwise
where c > 0.
(a) Find c.
(b) Find the cdf F ().
(c) Find the 50th percentile (the median) for the
distribution.
(d) Find the general formula for F^-1 (p), the 100pth percentile
of the distribution when 0 < p < 1.

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 be a continuous uniform (-2,5) random variable. Let W =
|X| Your goal is to find the pdf of W.
a)Begin by finding the sample space of W
b)Translate the following into a probability statement about X:
Fw(w) = P[W <= w] = ....
c) Consider different values of W the sample of W. Do you need
to break up the sample space into cases?
d)Find the cdf of W
e)Find the pdf of W

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Suppose that X is continuous random variable with PDF f(x) and
CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly
infinite) interval of the real numbers then F(x) is a strictly
increasing function of x over that interval. [Hint: Try proof by
contradiction]. (b) Under the conditions described in part (a),
find and identify the distribution of Y = F(x).

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