Question

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

Answer #1

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 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.

3. Let X be a continuous random variable with PDF
fX(x) = c / x^1/2, 0 < x < 1.
(a) Find the value of c such that fX(x) is indeed a PDF. Is this
PDF bounded?
(b) Determine and sketch the graph of the CDF of X.
(c) Compute each of the following:
(i) P(X > 0.5).
(ii) P(X = 0).
(ii) The median of X.
(ii) The mean of X.

A continuous random variable X has pdf ?x(?) = (? + 1) ?^2, 0 ≤
? ≤ ? + 1, Where B is the
last digit of your registration number (e.g. for FA18-BEE-123,
B=3).
a) Find the value of a
b) Find cumulative distribution function (CDF) of X i.e. ??
(?).
c) Find the mean of X
d) Find variance of X.

Let
the random variable X have pdf
f(x) = x^2/18; -3 < x < 3 and zero otherwise.
a) Find the pdf of Y= X^2
b) Find the CDF of Y= X^2
c) Find P(Y<1.9)

a continuous random variable X has a pdf f(x) = cx, for
1<x<4, and zero otherwise.
a. find c
b. find F(x)

The random variable X has the PDF
fX(x) = { 1/4 -3<=x<=1
{ 0 otherwise
If Y = (X - 2)^2 Find E|Y| Var|Y|

Let X be a random variable with cdf ?(?) = 0 x<1
1/2(x^2)-x+3/4 1<=x<2
1 x>=2
(a) (1 pt) Find the median of X
(b) Find the pdf f(x)
(c) (1 pts) Find the variance of X.

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 ?

Suppose the random variable X has pdf f(x;?, ?)=??x?−1e−?x? for
x≥0;?, ? > 0.
a) Find the maximum likelihood estimator for ?, assuming that ?
is known.
b) Suppose ? and ? are both unknown. Write down the equations
that would be solved simultaneously to ﬁnd the maximum likelihood
estimators of ? and ?.

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