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STT 231 – 001
E. Dikong
HOMEWORK 2
DUE DATE: WEDNESDAY APRIL 09, 2014. IF YOU SUBMIT HW #2 ON OR
BEFORE MONDAY APRIL 07, 2014, YOU EARN AN EXTRA BONUS OF 5
POINTS.
GROUP NUMBER : _________________________________________
LAST NAME
FIRST NAME
PID AND SIGNATURE
GROUP SCORE ________________________________________________________
1
1. Two cards are selected at random from a box which contains five cards numbered
1, 1, 2, 2, and 3. Let X denotes the sum and Y the maximum of the two numbers
drawn. Determine
(a) The joint distribution of X and Y. [Hint: Complete the table below] 13 points
Y
X
(b) Show clearly in the space provided below, your work for the P(X=2, Y=1). 2 points
(c) Show clearly in the space provided below, your work for the P(X=3, Y=2). 2 points
2
2. Suppose the distance X between a point target and a shot aimed at the point in a coinoperated target game is a continuous random variable with p.d.f.
3
2
 (1  x ),
f ( x)   4
0
1  x  1
otherwise.
(a) Sketch the graph of f(x). 4 points
(b) Compute P(X > 0). 3 points
3
(c) Compute P(0.5  X  0.5). 3 points
(d) Compute P( X  0.25 or X  0.25). 6 points
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3. A fair coin is tossed four times. Let X denotes the number of heads occurring and let Y
denote the longest string of heads occurring. Determine the joint distribution of X and Y.
[23 points] HINT: Complete the cells in the table below.
Y
X
5
4. The cook in a restaurant stashes away a tub containing 15 oysters because he knows
that there are pearls in 9 of the oysters. A busboy who also knows about the pearls finds
the tub, but can make off with only 4 of the oysters before someone sees him. If you let X
be the number of oysters that contain a pearl out of those the busboy has, [18 points]
(a) Write down the probability function of X? (4 points)
(b) What’s the probability that he has 0, 1, 2, 3, or 4 oysters with a pearl? [Hint:
Evaluate P(X=0); P(X=1); P(X=2); P(X=3); P(X=4).] (14 points) [Show work
and simplify your answers]
P(X = 0) =
P(X = 1) =
6
P(X = 2) =
P(X = 3) =
P(X = 4) =
(c) Find the E(X) and VAR(X). [4 points] [Work must be shown]
E(X) =
VAR(X) =
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5. Cathy has 6 blue, 3 yellow, 4 red, and 7 green marbles in a bag. She wants to give
her cousin Charlie 2 marbles of each color. If she reaches in the bag and takes out 8
marbles without looking and without replacing any, what’s the probability that she’ll
select 2 marbles of each color? (4 points) [Work must be shown and simplify your
answer]
6. Determine the value of c so that each of the following can serve as a probability
distribution function of the discrete random variable X. [HINT: Use  P( X  x)  1
where P(X = x) = f(x)] [Show work] (4 + 5) points
(i) f ( x)  c( x 2  4)
 2  3 

(ii) f ( x)  c 
 x  3  x 
for x  0, 1, 2, 3;
for x  0, 1, 2.
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7. A carnival game offers a $100 cash prize for anyone who can break a balloon by
throwing a dart at it. It costs $5 to play, and you are willing to spend up to $20 trying to
win. You estimate that you have about a 10% chance of hitting the balloon on any throw.
(i) Create a probability model for this carnival game [Hint: Give the model in the form of
a probability distribution table]. 10 points
(ii) Find the expected number of darts you will throw. 2 points
(iii) Find your expected winnings. 2 points
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8. Suppose the distance X between a point target and a shot aimed at the point in a coinoperated target game has the function f(x), where
 3(1  x 2 )
,  1  x  1,

f ( x)  
4
0,
otherwise.

Show that f(x) is a probability density function. 15 points
10
9. Let X be a binomially distributed random variable with E(X) = 2, and VAR(X) =
3
.
2
Find the distribution of X.
(i) First find the parameters p, q, k, and n. [Show clear work] 5 points
p=
q=
n=
k=
(ii) Give the distribution of X in the form of a table. [Do not simplify the binomial
expressions] 9 points
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10. (i) Suppose that a system contains a certain type of component whose time in years to
failure is given by the random variable T, distributed exponentially with parameter
  0.2 . If 5 of these components are installed in different systems, what is the
probability that at least 2 are still functioning at the end of 8 years? 10 points
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(ii) The length of time for one individual to be served at a cafeteria is a random variable
having an exponential distribution with a mean of 4 minutes. What is the probability that
a person is served in less than 3 minutes on at least 4 of the next 6 days? 10 points
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