Question

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.

(d) Find the expected value of X

Answer #1

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 ).

Consider a continuous random variable X with the probability
density function f X ( x ) = |x|/C , – 2 ≤ x ≤ 1, zero elsewhere.
a) Find the value of C that makes f X ( x ) a valid probability
density function. b) Find the cumulative distribution function of
X, F X ( x ).

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.

Suppose that X is continuous random variable with PDF f(x) and
CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly
infinite) interval of the real numbers then F(x) is a strictly
increasing function of x over that interval. [Hint: Try proof by
contradiction]. (b) Under the conditions described in part (a),
find and identify the distribution of Y = F(x).

X is a continuous random variable with the cumulative
distribution function
F(x) = 0
when x <
0
= x2
when 0 ≤ x ≤
1
= 1
when x >
1
Compute P(1/4 < X ≤ 1/2)
What is f(x), the probability density function of X?
Compute E[X]

7. For the random variable x with probability density function:
f(x) = {1/2 if 0 < x< 1, x − 1 if 1 ≤ x < 2}
a. (4 points) Find the CDF function. b. (3 points) Find p(x <
1.5). c. (3 points) Find P(X<0.5 or X>1.5)

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)

Suppose a random variable X has cumulative distribution function
(cdf) F and probability
density function (pdf) f. Consider the random variable Y =
X?b
a for a > 0 and real b.
(a) Let G(x) = P(Y x) denote the cdf of Y . What is the
relationship between the functions
G and F? Explain your answer clearly.
(b) Let g(x) denote the pdf of Y . How are the two functions f
and g related?
Note: Here, Y is...

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)=

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 .

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