Question

3. Suppose that a random variable X has the density function f (x) = 2x, 0 ≤ x ≤ 1, and that f (x) = 0 for x <0 and x > 1. a) Calculate the distribution function for X, as well as the corresponding E (X) and variance V (X). b) The numbers u1 = 0.0503, u2 = 0.9149, u3 = 0.3103, u4 = 0.1866, u5 = 0.6553 uniformly distributed, independent random numbers on the interval [0, 1]. Calculate, with justifications, five independent random numbers x1,. . . , x5 for the variable X. c) Calculate sample mean and sample variance for the simulated numbers for X. Compare with the result in (a). (2p)

Answer #1

A uniform random variable on (0,1), X, has density function f(x)
= 1, 0 < x < 1. Let Y = X1 + X2 where X1 and X2 are
independent and identically distributed uniform random variables on
(0,1).
1) By considering the cumulant generating function of Y ,
determine the first three cumulants 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.

A continuous random variable X has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise
(a) Find the value c such that f(x) is indeed
a density function.
(b) Write out the cumulative distribution function of
X.
(c) P(1 < X < 3) =?
(d) Write out the mean and variance of X.
(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of Y.

suppose x is a continuous random variable with probability
density function f(x)= (x^2)/9 if 0<x<3 0 otherwise
find the mean and variance of x

5. Consider the random variable X with the following
distribution function for a > 0, β > 0:
FX (z) = 0 for z ≤ 0
= 1 – exp [–(z/a)β] for z > 0 (where exp y = ey)
(a) Determine the inverse function of FX (z), where 0 < z
< 1.
(b) Let a = β = 2 for the random variable X, and define the
numbers u1 = .33 and u2 = .9. Use the inverse...

6. A continuous random variable X has probability density
function
f(x) =
0 if x< 0
x/4 if 0 < or = x< 2
1/2 if 2 < or = x< 3
0 if x> or = 3
(a) Find P(X<1)
(b) Find P(X<2.5)
(c) Find the cumulative distribution function F(x) = P(X< or
= x). Be sure to define the function for all real numbers x. (Hint:
The cdf will involve four pieces, depending on an interval/range
for 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)

A random variable X has the following pdf
f(x)=2x^-3, if x ≥1
0, Otherwise
(a) Find the cdf of X (b) Give a formula for the pth quantile of X
and use it to ﬁnd the median of X. (c) Find the mean and variance
of X

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1]
u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]
(a) Explain why these vectors cannot possibly be
independent.
(b) Form a matrix A whose columns are the ui’s and compute the
rref(A).
(c) Solve the homogeneous system Ax = 0 in parametric form and
then in vector form. (Be sure the...

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...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 8 minutes ago

asked 34 minutes ago

asked 46 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago