Question

2. Let X be a continuous random variable with pdf given by f(x) = k 6x − x 2 − 8 2 ≤ x ≤ 4; 0 otherwise.

(a) Find k.

(b) Find P(2.4 < X < 3.1).

(c) Determine the cumulative distribution function.

(d) Find the expected value of X.

(e) Find the variance of X

Answer #1

2. Let the probability density function (pdf) of random variable
X be given by:
f(x) = C (2x -
x²),
for
0< x < 2,
f(x) = 0,
otherwise
Find the value of
C.
(5points)
Find cumulative probability function
F(x)
(5points)
Find P (0 < X < 1), P (1< X < 2), P (2 < X
<3)
(3points)
Find the mean, : , and variance,
F².
(6points)

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 ).

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.

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)

Probability density function of the continuous random variable X
is given by f(x) = ( ce −1 8 x for x ≥ 0 0 elsewhere
(a) Determine the value of the constant c.
(b) Find P(X ≤ 36).
(c) Determine k such that P(X > k) = e −2 .

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).

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 X be a continuous random variable with a PDF of the form
fX(x)={c(1−x),0,if x∈[0,1],otherwise.
Find the following values.
1. c=
2. P(X=1/2)=
3. P(X∈{1/k:k integer, k≥2})=
4. P(X≤1/2)=

Let X be a continuous random variable with a PDF of the form
fX(x)={c(1−x),0,if x∈[0,1],otherwise.
c=
P(X=1/2)=
P(X∈{1/k:k integer, k≥2})=
P(X≤1/2)=

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.

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