Question

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.

Answer #1

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?

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.
c=
P(X=1/2)=
P(X∈{1/k:k integer, k≥2})=
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.
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 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.

1. Let X be a random variable with PDF f(x) = C*absolute
value(x), -1 <= x <= 1
A. Find the constant and plot the PDFof X. Identify P(X >
0.5) in the plot.
B. Determine and plot the CDF of X. Identify P(X > 0.5) in
the plot.
C. Compute E(X^2 + X + 1).

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 ? 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 ?

Let
X be a continuous random variable rv distributed via the pdf f(x)
=4e^(-4x) on the interval [0, infinity].
a) compute the cdf of X
b) compute E(X)
c) compute E(-2X)
d) compute E(X^2)

Let X be a continuous random variable with the following
probability density function:
f(x) = e^−(x−1) for x ≥ 1; 0 elsewhere
(i) Find P(0.5 < X < 2).
(ii) Find the value such that random variable X exceeds it 50%
of the time. This value is called the median of the random variable
X.

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