1. In a gambling game, a woman is paid $3 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should   she pay to play if the game is fair?

P(Jack or Queen) = 8/52

P(King or Ace) = 8/52

P(Any otℎer Card) = 36/52

E(X) = (8/52) ∗ $3 + (8/52) ∗ $5 + (36/52) ∗ $0 = $1.23

2. Let X be a random variable with density function

f(x) = { 

1 < x < 2,

elsewhere

Find the expected value of g(X) = 2X + 1.

E[g(X)] = ∫   13 (2x + 1)(x2  + 2x − 1)dx

3     2

1

=  3  [2 x4  + 5 x 3  − x]   = 109

3. Find E(Y/X) for the density function

f(x, y) = {                     0,

0 < x < 1, 0 < y < 1,

elsewhere

E [ ] = ∫  ∫   6  x 3  (1 − y2 ) dxdy

0      0

4. Let the random variable X represent the number of drives a server operates per workflow. The probability distribution for the number of drivers per sales workflow is

The probability distribution for the number of drivers per inventory workflow is

Show that the variance of the probability distribution for the sales workflow is smaller than that of the inventory workflow.

us   =  E(X)  =  (50)(0.3)  +  (75)(0.5)  +  (100)(0.2)  =  72.5

oS(2)  = (50 − 72.5)2 (0.3) + (75 − 72.5)2 (0.5) + (100 − 72.5)2 (0.2) = 306 .25

uI   =  E(X)  =  (50)(0.2)  +  (100)(0.5)  +  (200)(0.3)  =  120

oS(2)  = (50 − 120)2 (0.2) + (100 − 120)2 (0.5) + (200 − 120)2 (0.3) = 3100

5.  The weekly views (in 10,000’s) of an average entertainment YouTube channel is a continuous random variable X having the probability density

f(x) = { (2x − 1),

1 < x < 2,

elsewhere

Find the mean and variance of X.

u  =  E(X) =    ∫ x(2x − 1)dx =    ∫  (2x2  − x)dx =     [        −    x2 ]   =

E(X2) =    ∫ x2 (2x − 1)dx =    ∫  (2x3  − x2 )dx =     [        −    x 3 ]   =

o 2  = E(X2) − (u)2  =  − ()   =  −  =  ≈ 0.0764 

6. The fraction X of business analytics majors and the fraction Y of computer science majors who compete for a job are described by the joint density function

f(x, y) = {xy,

0 < x < 1, 1 < y < 2,

elsewhere

Find the covariance and correlation coefficient of X and Y.

g(x) = 3 ∫ xy dy = 6 x[y2 |1(2)] = 2x, 0 < x < 1

4    1                       4                     2

0

uX  = E(X) = 12x2 dx =

1                          1

0

2    2                       2                     14

1

E(Y2) = 6 ∫  y 3 dy = 6 (16 − 1) =  6 

E(XY) = 2 1x2y 2 dxdy =

oX(2)  =  − ()   =

15        14  2           13

oXY  = E(XY) − uX uY  =  − () () = 0

0

pXY   =   = 0

√() ()

7. If the joint density function of X and Y is given by

f(x, y) = { 

Find the expected value of g(X, Y) =  + X2 Y

E(g(X, Y)) =  ∫  ∫   dxdy =

8. A random variable X has a mean u = 10 and a variance G 2  = 4. Using Chebyshev’s theorem, find

(a)P (|X  −  10| ≥  3) =

(b)P (|X  −  10| <  3) =

(c)P(5  <  X  <  15) =

(a)P (|X  −  10| ≥  3) =

(b)P (|X  −  10| <  3) =

(c)P(5  <  X  <  15) =

9. It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are

inoculated, find the probability that

(a) none contracts the disease;

(b) fewer than 2 contract the disease;

(c) more than 3 contract the disease.

(a) p = 0.60; n = 5;x = 0; P(x = 0) = 0.01024

(b) p = 0.60; n = 5;x = 0,1; P(x < 2) = P(x = 0) + P(x = 1) = 0.01024 + 0.0768 = 0.0892  (c) p = 0.60; n = 5;x = 4, 5; P(x > 3) = P(x = 4) + P(x = 5) = 0.2592 + 0.07776 = 0.33696

10. A student drives to school and encounters a traffic signal. The traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school each weekday between 8:00 and 8:30 a.m. Let X1 be the number of times he encounters a green light, X2 be the number of times he encounters a yellow light, and X3 be the number of times he encounters a red light.     Find the joint distribution of X1, X2, and X3 .

n

x1 , x2 , x3

11. A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the prob- ability distribution of the random variable X representing the number of doctors on the committee. Find P(2 ≤ X ≤ 3).

Population Size = 6, Sample Size = 3, Success = 4

ℎ(x; 6,3,4) = (x(4))(3 x )  for x = 1,2,3

(3(6))       ,

P(2  ≤  X  ≤  3) =

12. From a lot of 10 missiles, 4 are selected at random and fired. If the lot contains 3 defective missiles

that will not fire, what is the probability that

(a) all 4 will fire?

(b) at most 2 will not fire?

Population Size = 10, Sample Size = 4, Success = 3; ℎ(x; 10,4,3) =  (a) ℎ(0; 10,4,3) =  =  =  =

(b) P(X ≤ 2) =  +  +  =  = 30

ℎ(0; 10,4,3) =  =  =  =

ℎ(1; 10,4,3) =  =  =  =  =

ℎ(2; 10,4,3) =  =  =  =  =