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This is the Solved Exam of Probability which includes Watched Gymnastics, Gymnastics and Baseball, Baseball and Soccer, Gymnastics And Soccer, Percentage, Primary Care Physician, Referral to a Specialist, Probability, Results etc. Key important points are: Different Experiments, Random, Average Time, Independent Random Variables, Independent Bernoulli, Sequence, Markov Chain, Determine, Weekend, Possible Value
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Final Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin
Problem Points Possible Points Earned 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10
Problem 1. An instructor gives her class a set of 20 problems with the information the final exam will consist of a random selection of 10 of them. If a student has figured out how to do 12 of the problems, what is the probability that he or she will correctly answer
(a) all 10 problems on the exam;
(b) at least 8 of the problems on the exam.
Solution. Imagine that an urn is filled with 20 balls, each representing one of the possible exam problems. Those balls corresponding to problems studied by the student are colored red; there are 12 such balls. The remaining balls are colored black. The instructor draws 10 of these balls at random without replacement. [Obviously, an instructor would not put the same problem twice on the exam! Thus, he samples without replacement.] Let X be the number of red balls drawn; this corresponds to the number of questions which the student can answer correctly.
(a)
P { X = 10} =
10
0
10
(b)
P { X ≥ 8} = P { X = 8} + P { X = 9} + P { X = 10} =
8
2
10
9
1
10
10
0
10
Problem 3. Urn A has 5 white and 7 black balls. Urn B has 3 white and 12 black balls. We flip a fair coin. If the outcome is heads, then a ball from urn A is selected, whereas if the outcome is tails, then a ball from urn B is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?
Solution.
3 15
1 2 5 12
1 2 +^
3 15
1 2 =
Problem 4. Suppose that X has the distribution function
F ( t ) =
0 if t < 1 1 − (^) t^1 3 if t ≥ 1.
Find E ( X ).
Solution. The density is given by differentiating F :
f ( t ) =
0 if t < 1 3 t −^4 if t > 1.
Thus,
E ( X ) =
−∞
t f ( t ) dt =
1
t (3 t −^4 ) dt =
− 3 t −^2 2
∞ 1
Problem 6. Let X and Y have joint pdf
f ( s , t ) =
se − s ( t^ +1)^ if s > 0, t > 0 , 0 otherwise.
(a) Find the conditional probability density function of Y given X = t.
(b) Find the density function of Z = X Y.
Solution.
fX ( s ) =
0
se − s ( t^ +1) dt = − e − s ( t^ +1)
∞ 0
= e − s^.
Consequently, if s > 0 and t > 0,
fY | X ( t | s ) =
f ( s , t ) fX ( s )
se − s ( t^ +1) e − s^
= se − st^.
That is, given X = s , the conditional distribution of Y is exponential with parameter s. We compute the density of Z in two ways. For u > 0
FZ ( u ) = P { X Y ≤ u } =
0
∫^ u / s
0
se − s ( t^ +1) dt ds =
0
e − s
∫ u / s
0
se − st^ dt ds
0
e − s^
− e − st^
] u / s 0 ds^ =
0
e − s^
1 − e − u^
ds =
1 − e − u^
Differentiating,
fZ ( u ) =
e − u^ if u ≥ 0 , 0 otherwise. We can also compute as follows:
P { X Y ≤ u } =
0
P { Y ≤ u / s | X = s } e − s^ ds =
0
(1 − e − u^ ) e − s^ ds = (1 − e − u^ )
0
e − s^ ds = (1 − e − u^ ).
The second inequality follows from part (a).
Problem 7. 12 people get on an elevator on the ground floor of a 10 story building. Each person selects a floor; assume that each person selects independently and each person picks one of the 10 possible floors uniformly at random. No new people get on the elevator after the ground floor. Compute the expected number of stops the elevator makes.
Solution. Let X be the number of stops the elevator makes. We can write X =
i = 1 Ii^ , where
Ii =
1 if the elevator stops at floor i , 0 otherwise.
Since expectation is linear, we have
i = 1
Ii
i = 1
E ( Ii ) =
i = 1
P {stop at floor i }.
Now
P {stop at floor i } = 1 − P {no-one picks floor i }
= 1 −
So
E ( X ) = 10
Problem 9. A column bet in roulette wins 2$ with probability 12/38, and losses 1$ with prob- ability 26/38. (a) Compute the mean and standard deviation of your winnings on a single game.
(b) You place this bet 25 times. Estimate the probability that you have won a positive amount.
(c) You place this bet 1000 times. Estimate the a probability that you have won a positive amount. Solution. Suppose that Xi is the amount won on the i th game. X 1 , X 2 ,... are independent and all have the same distribution. Then
E ( X 1 ) = 2
and E
Thus V ( X 1 ) = E
and so SD ( X 1 ) ≈ 1.394. Let Sn =
∑ n i = 1 Xi^. Write^ μ^ for^ E ( X^1 ) and^ σ^ for^ SD ( X^1 ).
P { S 25 > 0} = P
S 25 − 25 μ 5 σ
S 25 − 25 μ 5 σ
S 25 − 25 μ 5 σ
S 1000 − 1000 μ p 1000 σ
S 25 − 1000 μ p 1000 σ
Problem 10. Let X have a Gamma( α , λ ) distribution, and let Y be an independent Gamma( β , λ ) random variable. Let Z = X + Y.
(a) Find the MGF of Z. You can use Table 7.2.
(b) What is the distribution of Z?
Solution.
MZ ( t ) = MX ( t ) MY ( t )
=
λ λ − t
) α ( λ λ − t
) β
λ λ − t
) α + β
Using Table 7.2, we see that Z has a Gamma( α + β , λ ) distribution.
