Question

Suppose that X is a continuous random variable with a probability density function that is a positive constant on the interval [8,20], and is 0 otherwise.

**a.** What is the positive constant mentioned
above?

**b.** Calculate P(10?X?15).

**c.** Find an expression for the CDF FX(x).
Calculate the following values.

FX(7)=

FX(11)=

FX(30)=

Answer #1

Given:

f(x) = a, in the interval 8 x20

= 0, otherwise

The value of a is got by noting that the Total Probability = 1

i.e.,

i.e.,

a(20-8) = 1

So,

a = 1/12 = 0.0833

(b)

So, pdf of x is written as:

f(x) = 1/12, 8 x 20

= 0, otherwise

(c)

(i)

CDF of X is got by integrating f(x) from 8 to X as follows:

(ii)

since pdf f(x) is defined on in the interval [8,20].

(iii)

(iv)

F(30) = 1,

as f(x) is defined in the interval [8,20].

Let X be a continuous random variable with a probability density function
fX (x) = 2xI (0,1) (x) and let it be the function´
Y (x) = e^−x
a. Find the expression for the probability density function fY (y).
b. Find the domain of the probability density function fY (y).

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 ).

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

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.

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....

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).

Suppose a random variable X has a mixed distribution of the
interval [0,1). X has probability 0.5 at x = 0, X has the density
function fx(x) = x for 0 < x < 1, and X has no
density or probability elsewhere. Find and graph the CDF of X.

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).
c) Compute E(Y )

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 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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