Question

Given the following cumulative probability
function:

0 x < -5

.10 -5 <= x < 0

.40 0 <= x < 5

F(x)= .50 5 <= x < 10

.75 10 <= x < 15

1.0 X >= 15

a. P( 0 <x<10)

b. P( 5<x<10)

c. P(x< 10)

d. P(x>5)

e. P(x=7)

f. Calculate f (x) and draw F (x) and F (x)

g. Calculate E (x)

h. Calculate the variance of X

i. Calculate the expected g (x) = 5x-2

Answer #1

Given the following probability function:
.10 x = 5
f(x) = . 20 x = -2, 8, 10
.30 x = 6
A. Calculate f(x) and make an appropriate drawing of f(x) and
F(x) (15 pts)
B. P (6 < X < 8) (5 pts)
C. P (x > 7) (5 pts)
D. P (x < 8) (5 pts)
E. The average of X (5 pts)
F. The fashion of X (5 pts)
G. The variance of X (10 pts)

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)

Suppose that the random variable X has the following cumulative
probability distribution
X: 0 1. 2. 3. 4
F(X): 0.1 0.29. 0.49. 0.8. 1.0
Part 1: Find P open parentheses 1 less or equal than
x less or equal than 2 close parentheses
Part 2: Determine the density function f(x).
Part 3: Find E(X).
Part 4: Find Var(X).
Part 5: Suppose Y = 2X - 3, for all of X, determine
E(Y) and Var(Y)

The measurement error in estimating the fortune of a certain
self-declared billionaire, X, has the following cumulative
distribution function:
F(x) = 0 for x < -2
F(x)= .5 + .09375(4x −x3/3) for -2 ≤ x ≤ 2
F(x) = 1 for x > 2
a) Give the probability density function for X in the interval
-2, 2].
b) What is the probability that X > 1?
c) What is the probability that the billionaire actually has a
fortune less than...

The joint probability density function of x and y is given by
f(x,y)=(x+y)/8 0<x<2, 0<y<2 0 otherwise
calculate the variance of (x+y)/2

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....

Problem 4 The joint probability density
function of the random variables X, Y is given as
f(x,y)=8xy
if 0 ≤ y ≤ x ≤ 1, and 0 elsewhere.
Find the marginal probability density functions.
Problem 5 Find the expected values E
(X) and E (Y) for the density function given
in Problem 4.
Problem 7. Using information from problems 4
and 5, find
Cov(X,Y).

A random variable X has its probability function given by
x
0
1
2
3
4
f(x)
0.3c
0.1c
c
0.2c
0.4c
a) Find c and F(x), the cumulative distribution function for X
(for all real values of X).
b) Find the probabilities of the event X = 6 and the event X
>= 4.
c) Find P(1 < X <= 4) and P(1 < X <= 4 | X <=
3).

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...

The random variable X has probability density function:
f(x) =
ke^(−x) 0 ≤ x ≤ ln 2
0 otherwise
Part a: Determine the value of k.
Part b: Find F(x), the cumulative distribution function of X.
Part c: Find E[X].
Part d: Find the variance and standard deviation of X.
All work must be shown for this question. R-Studio should not be
used.

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