Question

Let X be a random variable with pdf given by fX(x) = Cx2(1−x)1(0
< x < 1), where C > 0 and 1(·) is the indicator
function.

(a) Find the value of the constant C such that fX is a valid
pdf.

(b) Find P(1/2 ≤ X < 1).

(c) Find P(X ≤ 1/2).

(d) Find P(X = 1/2).

(e) Find P(1 ≤ X ≤ 2).

(f) Find EX.

Answer #1

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.

5. Let X be a continuous random variable with PDF
fX(x)= c(2+x), −2 < x < −1,
c(2−x), 1<x<2,
0, elsewhere
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Determine the CDF of X and sketch its graph.
(c) Find P(X < 1.5).
(d) Find m = π0.5 of X. Is it unique?

A random variable X has the pdf given by fx(x) = cx^-3, x .>
2 with a constant c. Find
a) the value of c
b) the probability P(3<X<5)
c) the mean E(X)

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)

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

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 ).

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)=

1. Let (X,Y ) be a pair of random variables with joint pdf given
by f(x,y) = 1(0 < x < 1,0 < y < 1).
(a) Find P(X + Y ≤ 1).
(b) Find P(|X −Y|≤ 1/2).
(c) Find the joint cdf F(x,y) of (X,Y ) for all (x,y) ∈R×R.
(d) Find the marginal pdf fX of X. (e) Find the marginal pdf fY
of Y .
(f) Find the conditional pdf f(x|y) of X|Y = y for 0...

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 X be a random variable with probability density function
fX(x) given by fX(x) = c(4 − x ^2 ) for |x| ≤ 2 and zero
otherwise.
Evaluate the constant c, and compute the cumulative distribution
function.
Let X be the random variable. Compute the following
probabilities.
a. Prob(X < 1)
b. Prob(X > 1/2)
c. Prob(X < 1|X > 1/2).

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 12 minutes ago

asked 14 minutes ago

asked 21 minutes ago

asked 37 minutes ago

asked 44 minutes ago

asked 48 minutes ago

asked 52 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago