Question

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(4*x* −x^{3}/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 -2?

d) What is the probability that X > .5?

e) What is the probability that X > 1 given that X >
.5?

f) Calculate the 37th percentile of X.

g) What is the expected value of X?

h) What is the expected value of
*X*^{2}?

i) What is the variance of X?

j) What is the probability that X is more than 1 above its expected
value?

Answer #1

a)

pdf f(x)=(d/dx)F(x)=0.09375*(4-3x^{2})

b)P(X>1)
=1-P(X<1)=1-(0.5+0.09375*(4*1-1^{3}/3))=0.15625

c)P(X<-2)=0

d)P(X>0.5)=1-P(X<0.5)=1-(0.5+0.09375*(4*0.5-0.5^{3}/3))=0.316406

e)P(X>1|X>0.5)=P(X>1)/P(X>0.5)=0.15625/0.316406=0.493827

f)for 37 th percentile ; x=-0.35

g)E(X)=
xf(x) dx =
0.09375*(4x-x^{3}) dx
=0.09375*(4x^{2}/2-x^{4}/4)|^{2}_{-2}
=0

h)

E(X^{2})=
x^{2}f(x) dx =
0.09375*(4x^{2}-x^{4}) dx
=0.09375*(4x^{3}/3-x^{5}/5)|^{2}_{-2}
=0.8

i)

Var(X)=E(X^{2})-(E(X))^{2} =0.8

j)

P(X>1+0)=P(X>1)=0.15625

Let the probability density of X be given by f(x) = c(4x - 2x^2
), 0 < x < 2; 0, otherwise. a) What is the value of c? b)
What is the cumulative distribution function of X?
c) Find P(X<1|(1/2)<X<(3/2)).

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]

13. If X follows the following cumulative probability
distribution:
0 X≤5
0.10 (X-5) 5≤X≤7
F (X) 0.20 + 0.20 (X-7) 7≤X≤11
1 X≥11
a. Calculate a probability function f (X) (10 pts)
b. Calculate the expected value of X and the Variance of X. (15
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c. Calculate the probability that X is between 6.0 and 8.80. (10
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d. Calculate the percentile of 70 percent. (10 pts)
e. Calculate the expected g (x), if g (X) = 2X-10 (15...

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3. Find a value for k such that f(x) = kx on the interval [2, 3]
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b. . Find P(0 ≤...

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a) Find the value of C that makes f X ( x ) a valid probability
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f(x)=Cx
1. what value should C be for this to be a valid probability
density function on the interval [0,4]?
2. what is the Cumulative distribution function f(x) which gives
P(X ≤ x) and use it to determine P(X ≤ 2).
3. what is the expected value of X?
4. figure out the value of E[6X+1] and Var(6X+1)

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
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Suppose that the random variable X has the following cumulative
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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).
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1. f is a probability density function for the random
variable X defined on the given interval. Find the
indicated probabilities.
f(x) = 1/36(9 − x2); [−3, 3]
(a) P(−1 ≤ X ≤ 1)(9 −
x2); [−3, 3]
(b) P(X ≤ 0)
(c) P(X > −1)
(d) P(X = 0)
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f(x) = kx2; [0,
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k=

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, if 0
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Find the expected value of X, E(X).
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