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MATH 19 B FINAL EXAM PROBABILITY REVIEW PROBLEMS SPRING, 2010
This handout is meant to provide a collection of exercises that use the material from the probability and statistics portion of the course. The answers to the exercises are at the end. Don’t take the exercises as indicative of the final exam problems. In particular, some of these exercises are quite involved and are designed not so much to test your knowledge as to broaden your view of a given topic as you work through the answers. In any event, if you can come to terms with the exercises below, you will do fine with the probability and statistics part of the final exam.
1 Which of the following are not probability functions on^ [0, 1]? (a) π 2 sin( πx ) (b) e − x^ (c) π 2 cos( πx ) (d) (^) ( x +^21 ) 2
2 Using the probability function e − x^ on [0, ∞), compute the probability that x lies in [1, 2] ∪ [3, ∞).
3 Using the uniform probability function^21 π on^ [0, 2 π ], compute the probability that sin^2 ( θ^ )^ is less than 1
4 Give the sample space^ [0,^ ∞)^ the exponential probability function^ e − x^. Suppose that^ a^ and^ b^ are such that 0 ≤ a < b. Write down the probability that the random variable x → x^2 has a value between a and b.
5 Given that x is less than 4, what is the probability as computed using the exponential function e − x^ on [0, ∞) that x is greater than 2?
6 Take the Gaussian probability function with mean 3 and standard deviation 2 for the line,^ (−∞,^ ∞). Are there sets A and B in (−∞, ∞) such that P ( A | B ) = 13 , P ( B | A ) = 101 , and P( B ) = 12?
7 Suppose that N is a positive integer. Write down the sample space for the possible outcomes of flipping N coins simultaneously. Write a sum that gives the probability of at least three heads on the N coins if the probability of getting heads on any one coin is 14 and if it is assumed that the outcome on any one flip has no bearing on that of any other.
8 Use the exponential function^ e − x^ as a probability function on^ [0,^ ∞).
(a) What is the probability that the closest integer to x is odd? (b) What is the conditional probability that the integer closest to x is odd given that x is larger than 1? (c) Is the event that the integer closest to x is odd independent from the event that x is greater than 2?
9 Compute the mean and standard deviation of the random variable x → x^2 for the probability function 6 x ( x^2 + 1 )^4 on^ [0,^ ∞).
10 Suppose that^ −∞ ≤^ a^ <^ b^ ≤ ∞^ and that^ p ( x )^ is a probability function on^ [ a ,^ b ]^ with mean^ μ. Explain why
∫ (^) b a x
(^2) p ( x ) d x − μ (^2) is the square of the standard deviation.
11 Use the probability function^12 cos( x )^ on^ [−^ π 2 ,^ π 2 ].^ What are the mean and standard deviation for the random variable x → sin^2 ( x )?
12 Use the probability function 12 cos( x ) on [− π 2 , π 2 ]. Which of the following is the probability that the random variable sin^2 ( x ) has value between 0 and 15?
(a)
1 5 (b)^
1 5 (c)^
π 5
1 5 (d)^
1 15
1 5 (e)^
1 15
MATH 19 B FINAL EXAM PROBABILITY REVIEW PROBLEMS SPRING, 2010
13 Let N be a positive integer, and let S denote the sample space for the possible outcomes of N coin flips. Give S the probability function for which the probability of any given coin landing heads is independent from any other, and for which the probability of any given coin landing heads is 14. Define f to be the random variable that multiplies the number of heads times the number of tails.
(a) Which is of the following is the mean of f : N , 14 ( N^2 + N ), 163 N^2 , or 163 N ( N − 1 )? (b) Assume that N > 2 and is even. Is f independent of the random variable that gives 0 when the number of heads is odd and 1 when the number of heads is even? (c) Write down a sum that gives the characteristic function for the random variable that is 0 when the number of heads is odd and 1 when the number of heads is even.
14 Use the probability function (^21) π on [− π , π ]. Are the random variables x → cos( x ) and x → sin( x ) independent?
15 Suppose that the probability that I am sick on any particular day is 1001 , the probability that I am absent any particular day is 2001 , and the probability that I am absent given that I am sick is 101. If I am absent on a given day, what is the probability that my absence is due to being sick?
16 Suppose that a system has four possible states, these labeled as^ {1, 2, 3, 4}. Suppose, in addition, that after any given unit of time, the probability from going from any even state to any odd state is 16 , the probability of going from any even state to any even state is 13 , the probability for going from any odd state to any even state is 13 and the probability from going from any odd state to any odd state is 16. Let ~p ( t ) denote the vector whose k th component is the probability of being in state k at time t.
(a) Write down a matrix, A , such that the equation ~p ( t + 1 ) = A~p ( t ) holds. (b) What is lim t →∞ ~p ( t )? (c) Denote the vector you found in part (b) by ~q. Explain why ~p ( 1 ) = ~q as long as the entries of ~p ( 0 ) sum to one.
17 Suppose that three coins are flipped simultaneously and we don’t know the probability of heads although we do know that the probability is the same for each coin, and that the probability for any one coin is independent of that for any other. Define a function, f , on the sample space to be 1 if there is an odd number of heads and zero otherwise. Thus, f has two possible values.
(a) Given that the probability of heads on any coin is some number p ∈ [0, 1], write down the probability that f = 1 in terms of p. (b) In terms of p , what is the mean and standard deviation for f? (c) Suppose that we now flip the three coins some large number of times and see that the average value of f is 167. According to the Central Limit Theorem, which of the following is the choice for p : 161 , 18 , 14 , 12? (d) If the fraction of times f is 1 after a large number of flips is 167 , what would a Bayesian guess for the probability function on the sample space for flipping three coins?
18 Suppose that an experiment is repeated 100 times and a certain measurement can have one of two values either 1 and 0 each time. In this regard, assume that the probability of measuring 1 is the same for each run, and that the probability for a measurement on any given run is independent of that for any other. If the probability of measuring 1 on any given run is postulated to be 101 , what is the probability of precisely n ∈ {0, 1,... , 100} of the runs giving the measurement 1?
MATH 19 B FINAL EXAM PROBABILITY REVIEW PROBLEMS SPRING, 2010
(b) Use the Chebychev Theorem to estimate an upper bound for the probability of seeing less than 144 wins in 900 games under the assumption that the dice are fair. You needn’t evaluate square roots or exponentials that appear in your answer.
28 Let S denote the sample space of pairs of the form ( j , k ) where j and k are integers in the set {1,... , n }. Suppose that S has a probability function, and let P
j | k
denote the conditional probability that the first entry in a pair is j given that the second entry is k.
(a) Explain why the matrix whose entry in the j th column and k th row is P
j | k
is a Markov matrix. (b) Suppose that A is an n × n Markov matrix and p is a probability function on the set {1,... , n }. Use A and p to determine a probability function on S with the following two properties: First, p ( k ) is the probability that the second entry of any given pair from S is k. Second, the conditional probability P
j | k
is Ajk.
29 Let^ A^ denote the Markov matrix
1 3
1 2 2 3
1 2
. Suppose that ~p ( 0 ) =
ñ 0 1
ô and that for any t ∈ {1, 2,... }, the
vector ~p ( t ) is defined to equal A~p ( t − 1 ).
(a) What is lim t →∞ ~p ( t )?
(b) How big must t be before before the difference between this limit and ~p ( t ) is a vector whose length is less than 1001?
30 Suppose that the concentration in the blood of a given medicinal drug is measured as a function of time. In particular, suppose that measurements are made at a sequence of times t 1 < t 2 < · · · < tn , that yield the corresponding sequence of values { y 1 ,... , yn }. In addition, suppose that it is believed that the function, t → y ( t ), that describes the concentration as a function of t has the form y ( t ) = ae − t^ + be −^2 t where a and b are constants. Derive an expression in terms of the data {( tk , yk )} for the pair ( a , b ) that minimizes
k (^ y ( tk )^ −^ yk )
31 Let S = {0, 1, 2,... , 9} and let p denote the probability function on S that assigns the value 101 to each digit.
(a) What are the mean and standard deviation for p? (b) What is the sample space for a sequence of 100 digits, each chosen from S? (c) Write down a Gaussian probability function that approximates the probabilities for the average 100 digits chosen at random from S. The integral of the latter function over any given interval of the form [ a , b ] should give a good approximation for the probability that the average of the 100 should be greater than or equal to a and less than or equal to b. Note that the term “at random” in this case should be taken to mean that any one of the 10^100 possible sequence of 100 digits has probability 10−^100 of appearing.
32 Suppose that on average, 1 case of a certain sort of cancer will appear per year in any population of 10, 000 individuals. Suppose a town of population 100, 000 sees 20 cases one year. Write an infinite sum that gives the P -value for the hypothesis that these cases are unrelated and that the appearance of this many cases is due to chance?
33 Suppose that 20% of people with a certain genetic variant develop a certain sort of skin cancer, and that 10% of the population of certain locale has this gene. Meanwhile, 5% of the people in this locale develop the cancer. Given that an individual from this locale has the cancer, what is the probability that the person has the genetic variant?
MATH 19 B FINAL EXAM PROBABILITY REVIEW PROBLEMS SPRING, 2010
34 Suppose that a certain sort of bacteria moves along a line by sequence of flips, with each flip either one body length in the + direction, or one body length in the − direction. Suppose that the rules for deciding are as follows: The probability of flipping in a given direction if the previous flip was in that direction is 2 3 ; and the probability of flipping in a given direction if the previous flip was in the opposite direction is 1
- For each^ t^ ∈ {1, 2,...^ }, let^ p +( t )^ denote the probability that the^ t th flip is in the^ +^ direction, and let p −( t ) denote the probability that the t th flip is in the minus direction.
(a) For each t > 1, give a linear equation that relates p +( t ) with p +( t − 1 ) and p −( t − 1 ). Then, do the same for p −( t ). (b) Let ~p ( t ) ∈ R^2 denote the vector whose top component is p +( t ) and whose bottom component is p −( t ). Find a matrix A so that the equation ~p ( t ) = A~p ( t − 1 ) holds for each t > 1. (c) Is A a Markov matrix? (d) Find the eigenvectors and eigenvalues of the matrix A. (e) Find ~p ( t ) given that p +( 1 ) = 1 and p −( 1 ) = 0. (f) Find lim t →∞ p +( t ).
35 Suppose that a survey of finds the following fact: When N is sufficiently large, roughly 51% of N births are female and so 49% are male. Said differently, (^) N^1 (#female − #male) = 0.02 when N is large. Granted this data, suppose that we make the hypothesis that the probability of having a female child is 12 and so the probability of having a male child is also 12.
(a) Under the assumption that the hypothesis of equal probabilities is correct, use the Central Limit Theorem to write down a probability function that gives an accurate approximation when N is large for the probability that 12 (#female − #male) has its value in any given interval. (b) Use the Chebychev theorem with the mean and standard deviation from your probability function from part (a) to answer the following question: How big must N be before our hypothesis of equal probabilities has P -value less than 0.05?
MATH 19 B PROBABILITY REVIEW PROBLEMS – ANSWERS SPRING, 2010
(b) When n is even, then f is even; and when n is odd, then f is odd. Thus, the random variable that gives 0 when n is odd and 1 when n is even can be written as ( 1 + (− 1 ) f^ ). This shows that the two random variables can not be independent. (c) Let α =
k is odd and 1 ≤ k ≤ N
N
k
( 14 ) k ( 34 ) N^ − k , thus α is the probability that n is odd.
The characteristic functional is equal to ( 1 − α ) + αt. Note that the sum for α can be computed in closed form: α = 12 + ( 12 ) N^ +^1.
14 These two random variables are not independent. This can be seen from the fact that the absolute value of one can be deduced knowing that of the other.
15 15. Use Bayes’ rule: P^ (Sick^ |^ Absent) =^ P(Absent P(^ Absent|^ Sick)^ )P (Sick).
16 (a)
1 6
1 6
1 6
1 6 1 3
1 3
1 3
1 3 1 6
1 6
1 6
1 6 1 3
1 3
1 3
1 3
(b) ~q =
1 6 1 3 1 6 1 3
(c) Since the span of the image of A is 1-dimensional (all of its columns are the same), A has three eigenvalues that are zero. Denoting these by ~v 1 , ~v 2 and ~v 3 , the vector ~p ( 0 ) must have the form ~q + a 1 ~v 1 + a 2 ~v 2 + a 3 ~v 3 since the entries of each ~vk sum to zero and the entries of ~q sum to 1. Thus, A~p ( 0 ) = ~q.
17 (a) p^3 + 3 p ( 1 − p )^2 = p ( 4 p^2 − 6 p + 3 ).
(b) μ = p ( 4 p^2 − 6 p + 3 ) and σ =
p μ − μ^2. (c) (^14) (d) P( H H H ) = P( H T T ) = P( T H T ) = P( T T H ) = 647 , P( T T T ) = P( T H H ) = P( H T H ) = P( H H T ) = 647.
To explain, the task is to guess the values for a probability function, P, on the set { H H H , H H T , H T H , T H H , H T T , T H T , T T H , T T T }. Label these elements from 1 to 8. Let q 0 = 169 and q 1 = 167. Then one can write q 0 = p 5 + p 6 + p 7 + p 8 and likewise write q 1 = p 1 + p 2 + p 3 + p 4.
This has the form of qa =
∑^8
j = 1
Ba j pj. The Bayesian will take pj =
a =0,
∑^ Ba j k Bak
qa.
n
( 101 ) n ( 109 )^100 − n^ = (^) n! ( 100 100!− n )! ( 101 ) n ( 109 )^100 − n.
19 The characteristic function is the polynomial in t given by t → P ( t ) = ( 109 + 101 t )^100. This is
equal to
n = 0
tn^ P( n ) where P( n ) =
n
( 101 ) n ( 109 )^100 − n^ is the probability of seeing n occurrences of 1. Differentiating the identity P ( t ) =
n
tn^ P( n ) three times finds that its third derivative at t = 1 equals
MATH 19 B PROBABILITY REVIEW PROBLEMS – ANSWERS SPRING, 2010
n ≥ 3
n ( n − 1 )( n − 2 ) P( n ). This is the mean of the random variable f ( n ). Meanwhile, differentiating
( 109 + 101 t )^100 three times and setting t = 1 gives 100 · 99 · 98 · 1013 = 88.2. Thus, the mean is 88.2.
20 Yes.^ The mean for the binomial probability in this case is 10 and the standard deviation is 3.^ As 25 − 10 = 15 = 5 σ , and the square of 5 is 25, the Chebychev inequality asserts that the probability of 25 or more occurrences of 1 is less than 251.
21 The average number of large meteor hits on the earth is 1 per 30 million years. Assuming that these meteors are not traveling together (that is, not two halves of some broken comet), how likely is it for a two huge meteors to hit the earth in 2005?
22 The mean is 25 so the standard deviation is 5. Granted this, use the Chebychev inequality.
23 (a) 1 − e − x^ , where x = 3 × 10 −^5. This is pretty close to 3 × 10 −^5. (b) 1 − e − x^ , where x = 3 × 10 −^5.
24 (a)^2 p^1 π e −( x −^10 )
(^2) / 8
(b)
6
1 2 p π e
−( x − 10 )^2 / (^8) d x
q 1000 3 π e
− 2000 ( x −7.5)^2 / 3
q 1000 3 π e
− 2000 ( x −7.5)^2 / 3
27 (a)
N
n
( 29 ) n ( 79 ) N^ − n^ = (^) n! ( NN −! n )! ( 29 ) n ( 79 ) N^ − n (b) Let S = { W , L } where W means win and L means lose. Define a random variable, f , on S so that f ( W ) = 1 and f ( L ) = 0. The mean of f is 29 and the standard deviation is
p 2 3.^ Let { f 1 ,... , f 900 } denote 900 identical versions of this same random variable. According to the Central Limit Theorem, the probabilities for the values of f = 9001 ( f 1 + · · · + f 900 ) are determined by the Gaussian probability function with mean 29 and standard deviation p 9001 ·
p 2 3 =
p 2
- Now,^
144 900 differs from 29 by 251 and this is R ·
p 2 90 when^ R^ =^
18 5 p
- The Central Limit Theorem therefore finds that the probability of less than 144 wins is less than p^1 π^95 e −^81 /^25.
28 (a) Each P^
j | k
is non-negative since these are probabilities, and P ( 1 | k ) + · · · + P ( n | k ) = 1 for each k since the sum of the conditional probabilities must be 1. (b) The definition of the conditional probability tells us that we should set P
j | k
to equal the quotient of probabilities P(( p^ ( j , kk ))). Thus, P(( j , k )) = P
j | k
p ( k ) = Ajk p ( k ).
29 (a) This limit is ~v 1 =
3 7 4 7
(b) The other eigenvalue is − 16 and the eigenvector is ~v 2 =
ñ 1 − 1
ô
. Thus, ~p ( 0 ) = ~v 1 − 37 ~v 2 and
~p ( t ) − ~v 1 = − 37 (− 16 ) t^ ~v 2 has norm
p 2 · 37 ( 16 ) t^. Thus, we need t ≥ (^) ln 6^1 ln( (^3007)
p 2 ) ≈ 2.3.
