Question 1 Suppose the following clothes are available in Sheila’s closet:
tops
bottoms
red jumper, blue blouse, green T-shirt
black trousers, brown skirt
Sheila chooses one item from the tops and one from the bottoms. How many outcomes
are in the sample space?
1. 5
2. 6
3. 7
Question 2 Suppose the following clothes are available in Sheila’s closet:
tops
bottoms
red jumper, blue blouse, green T-shirt
black trousers, brown skirt
Sheila chooses one item from the tops and one from the bottoms. Let A be the event
that Sheila chooses the brown skirt. How many outcomes are in A?
1. 2
2. 3
3. 6
Question 3 A medical test for a particular disease detects whether a patient is either
healthy or has the disease. The proportions in the general population are: 90% are
healthy, 7% have a mild form of the disease and 3% have a strong form of the disease. For healthy patients the test gives a false positive result with probability 0.1. For
patients with the mild form of disease the test gives a positive result with probability
0.95 and for patients with the strong form it always gives a positive result. What is the
probability that a randomly selected person is healthy and has a positive test result?
1. 0.1
2. 0.1 × 0.9
3. 0.1 × 0.9 + 0.07 × 0.95 + 0.03 × 1
Question 4 A medical test for a particular disease detects whether a patient is either
healthy or has the disease. The proportions in the general population are: 90% are
healthy, 7% have a mild form of the disease and 3% have a strong form of the disease. For healthy patients the test gives a false positive result with probability 0.1. For
patients with the mild form of disease the test gives a positive result with probability
0.95 and for patients with the strong form it always gives a positive result. What is the
probability that a randomly selected person has a positive test result?
1. 0.1 × 0.95
1
2. 0.1 × 0.9
3. 0.1 × 0.9 + 0.07 × 0.95 + 0.03 × 1
Question 5 A medical test for a particular disease detects whether a patient is either
healthy or has the disease. The proportions in the general population are: 90% are
healthy, 7% have a mild form of the disease and 3% have a strong form of the disease. For healthy patients the test gives a false positive result with probability 0.1. For
patients with the mild form of disease the test gives a positive result with probability
0.95 and for patients with the strong form it always gives a positive result. What is the
conditional probability that a person is healthy given he has a positive test result?
1. 0.1 × 0.9
2.
0.1×0.9
0.1×0.9+0.07×0.95+0.03×1
3. 0.1 × 0.9 + 0.07 × 0.95 + 0.03 × 1
Question 6 A factory has three production lines L1 , L2 , and L3 . The lines produce defective products with probabilities 3%, 2% and 1% respectively. Line L1 produces 10%
of all products, L2 30% and L3 60%. Given a faulty product, what is the conditional
probability that it was produced on line L1 ?
1.
0.03×0.1
0.03×0.1+0.02×0.3+0.01×0.6
2. 0.03 × 0.01
3.
0.03×0.01
0.1
Question 7 A factory has three production lines L1 , L2 , and L3 . The lines produce
defective products with probabilities 3%, 2% and 1% respectively. Line L1 produces
10% of all products, L2 30% and L3 60%. Given a non-faulty product, what is the
conditional probability that it was produced on line L3 ?
1. 0.01 × 0.6
2.
0.01×0.6
0.03×0.1+0.02×0.3+0.01×0.6
3.
0.99×0.6
0.97×0.1+0.98×0.3+0.99×0.6
Question 8 Suppose we flip 4 coins independently, each coming up heads with probability p. What is the probability of having an equal number of heads and tails?
1. p2 (1 − p)2
2. 4 × p2 (1 − p)2
3. 6 × p2 (1 − p)2
Question 9 A machine produces defective items with probability 0.1. Whether an item
is defective is independent of the quality of any other item. What is the probability that
in a sample of three, at most one item is defective?
2
1. 0.1 × 0.92
3
2.
0.1 × 0.92
1
3. 3 × 0.1 × 0.92 + 0.93
Question 10 A machine produces defective items with probability 0.1. Whether an
item is defective is independent of the quality of any other item. If you start the machine
in the morning, what is the probability that the 15th item is the first defective one?
1. 0.1 × 0.914
15
2.
0.1 × 0.914
1
3. 0.1 × 0.915
Question 11 For a ≤ x ≤ b what is the cumulative distribution function F (x) of a
random variable X uniformly distributed over the interval (a, b)?
1.
1
b−a
2.
x−a
b−a
3.
x
b−a
Question 12 Suppose X is a N (µ, σ 2 ) random variable. What is E(X 2 )?
1. 0
2. µ2
3. σ 2 + µ2
Question 13 Consider a multiple choice exam with two possible answers for each of
five questions. What is the probability that a students would get four or more correct
answers just by guessing?
1.
14
2
+
15
2
2. 4 ×
14
2
+
15
2
3. 5 ×
15
2
+
15
2
Question 14 Let X be a random variable with probability density
3
2
4 (1 − x ) −1 < x < 1
f (x) =
0
otherwise
For −1 < x < 1 what is the (cumulative) distribution function F (x) of X?
3
1.
1
4 x(3
2.
3
4 (1
− x)
3.
1
4 (1
− x2 )3
− x2 ) +
1
2
Question 15 Let X be a random variable with probability density
3
2
4 (1 − x ) −1 < x < 1
f (x) =
0
otherwise
What is the expectation of X?
1. −1
2.
1
4
3. 0
Question 16 Let X be a random variable with probability density
3
2
4 (1 − x ) −1 < x < 1
f (x) =
0
otherwise
What is E(X 2 )?
1. 0
2. 2/5
3. 1/5
Question 17 Suppose X is a random variable with E(X) = 1 and Var(X) = 9. Let
Y be a random variable with E(Y ) = 3, Var(Y) = 25 and Cov(X, Y) = −2. Define
W = X − Y . What is Var(W)?
1. 25
2. 30
3. 38
Question 18 Suppose X is a random variable with E(X) = 1 and Var(X) = 9. What
is E((X + 2)2 )?
1. 18
2. 12
3. 14
Question 19 Suppose X is a random variable with E(X) = 2 and Var(X) = 5. What
is E((X + 1)2 )?
4
1. 10
2. 12
3. 14
Question 20 Suppose X is a random variable with E(X) = 1 and Var(X) = 9. What
is Var(4 + 3X)?
1. 81
2. 85
3. 96
Question 21 Suppose X is a random variable with E(X) = 1 and Var(X) = 9. Let
Y be a random variable with E(Y ) = 3, Var(Y) = 25 and Cov(X, Y) = −2. Define
W = X − 2Y . What is E(W )?
1. 5
2. 0
3. -5
Question 22 Suppose X is a random variable with E(X) = 1 and Var(X) = 9. Let
Y be a random variable with E(Y ) = 3, Var(Y) = 25 and Cov(X, Y) = −2. Define
W = X − 2Y . What is Var(W)?
1. 109
2. 101
3. 117
Question 23 Suppose X is a random variable with E(X) = 1 and Var(X) = 9. Let
Y be a random variable with E(Y ) = 3, Var(Y) = 25 and Cov(X, Y) = −2. Define
W = X − 2Y . What is Cov(X, W)?
1. 0
2. 13
3. 5
Question 24 If X is a non-negative random variable with cumulative distribution
function F (x). What is the cumulative distribution function of log(X)?
1. log(F (x))
2. exp(F (x))
3. F (exp(x))
5
Question 25 If X is a non-negative random variable with cumulative distribution
function F (x). What is the cumulative distribution function of exp(X)?
1. log(F (x))
2. F (log(x))
3. F (exp(x))
Question 26 If X is a non-negative random variable with cumulative distribution
function F (x). What is the cumulative distribution function of X 2 ?
√
1. F ( x)
2. F (x2 )
3. F (x)2
Question 27 Suppose the random variable X counts the number of goals a footballer
scores in 3 attempts. If P(X = 1) = P(X = 2) = 0.3 and P(X = 3) = 3×P(X = 0),
what is E(X)?
1. 1.5
2. 1.8
3. 2.0
Question 28 Suppose the random variable X counts the number of goals a footballer
scores in 3 attempts. If P(X = 1) = P(X = 2) = 0.3 and P(X = 3) = 3×P(X = 0),
what is Var(X)?
1. 1.5
2. 1.8
3. 0.96
Question 29 Suppose the random variable x counts the number of goals a footballer
scores in 3 attempts. If P(X = 1) = P(X = 2) = 0.3 and P(X = 3) = 3×P(X = 0),
what is E(X 2 )?
1. 4.2
2. 5.0
3. 10.0
Question 30 Suppose the state space of the random variable X is S = {0, 1, 2}. If
P(X = i) = 12 P(X = i − 1) for i = 1, 2 what is E(X)?
1.
3
7
6
2.
4
7
3.
1
2
Question 31 If X is continuous with distribution function F and density f . What is
the density of Y = 2X?
1. f (2y)
2. f (y/2)
3.
1
2 f (y/2)
Question 32 Suppose the length of a phone call (in minutes) is exponentially distributed with parameter λ = 1/5. Your friend makes a phone call. What is the probability that you have to wait between 5 minutes and 10 minutes?
1. e−1 + e−2
2. 1 − e−1
3. e−1 − e−2
Question 33 Suppose the length of a phone call (in minutes) is exponentially distributed with parameter λ = 1/5. Your friend makes a phone call. What is the probability that you have to wait more than 5 minutes?
1. e−1
2. 1 − e−1
3.
1 −1
5e
Question 34 If X has an exponential distribution with parameter λ. Compute the
distribution function of X/λ. Which distribution does X/λ have?
1. Uniform(0,1)
2. Exponential(λ2 )
3. Exponential(1)
Question 35 Let X be a continuous random variable. Set Y = F (X) and compute
the distribution function of Y . Which distribution does Y have?
1. Standard Normal
2. Uniform(0,1)
3. the same distribution as X
7
Question 36 Let X and Y have joint density function
3 2
2 x (1 − |y|) for − 1 < x, y < 1
f (x, y) =
0
otherwise
Let A = {(x, y) : 0 < x < 1, 0 < y < 1}. What is the probability that (X, Y ) ∈ A?
1.
1
4
2.
1
2
3.
3
4
Question 37 Let X and Y have joint density function
3 2
2 x (1 − |y|) for − 1 < x < 1, −1 < y < 1
f (x, y) =
0
otherwise
Let A = {(x, y) : 0 < x < 1, 0 < y < x}. What is the probability that (X, Y ) ∈ A?
1.
9
40
2.
1
2
3.
3
4
Question 38 Let X and Y have joint density function
3 2
2 x (1 − |y|) for − 1 < x, y < 1
f (x, y) =
0
otherwise
What is the marginal pdf of Y ?
1. (1 − y 2 )
2.
3
2 (1
− |y|)
3. (1 − |y|)
Question 39 Let X and Y have joint density function
3 2
2 x (1 − |y|) for − 1 < x, y < 1
f (x, y) =
0
otherwise
What is the marginal pdf of X?
1. x2
2.
3
2 (1
3.
3 2
2x
− |x|)
8
Question 40 Let X and Y have joint density function
3 2
2 x (1 − |y|) for − 1 < x, y < 1
f (x, y) =
0
otherwise
Are X and Y independent?
1. Yes
2. No
Question 41 Let X and Y have joint density function
2 for 0 < x < y < 1
f (x, y) =
0 otherwise
What is the marginal pdf of Y ?
1. y
2. x/y
3. 2y
Question 42 Let X and Y have joint density function
2 for 0 < x < y < 1
f (x, y) =
0 otherwise
What is the marginal pdf of X?
1. 2x
2. 2(1 − x)
3. (1 − x)
Question 43 Let X and Y have joint density function
2 for 0 < x < y < 1
f (x, y) =
0 otherwise
Are X and Y independent?
1. Yes
2. No
Question 44 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is the marginal pmf of X?
9
1.
2x+5
16
2.
2x
32
3.
x
32
Question 45 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is the marginal pmf of Y ?
1.
2y+5
16
2.
2y+3
32
3.
y
32
Question 46 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is P(X ≥ Y + 1)?
1.
3
16
2.
9
32
3.
3
32
Question 47 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is P(Y = 2X − 1)?
1.
3
16
2.
7
32
3.
3
32
Question 48 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is P(X + Y = 4)?
10
1.
8
32
2.
9
32
3.
10
32
Question 49 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is P(X ≤ 3 − Y )?
1.
2
9
2.
4
32
3.
1
4
Question 50 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
Are X and Y independent?
1. Yes
2. No
Question 51 Let X and Y have joint density function
2 for 0 < x < y < 1
f (x, y) =
0 otherwise
What is the conditional density function of X given Y = 0.5?
2 if 0 < x < 0.5
1. f (x|y = 0.5) =
0 otherwise
(1 − x) if 0 < x < 0.5
2. f (x|y = 0.5) =
0
otherwise
0.5 if 0 < x < y
3. f (x|y = 0.5) =
0
otherwise
Question 52 Let X and Y have joint density function
2 for 0 < x < y < 1
f (x, y) =
0 otherwise
What is the conditional density function of Y given X = 0.5?
11
y
0
if 0.5 < y < 1
otherwise
2
0
if 0.5 < y < 1
otherwise
2
0
if 0 < y < 1
otherwise
1. f (y|x = 0.5) =
2. f (y|x = 0.5) =
3. f (y|x = 0.5) =
Question 53 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is the conditional pmf of X given Y = 2?
1. p(x|y = 2) =
x+2
4
2. p(x|y = 2) =
x
32
3. p(x|y = 2) =
x+2
7
for x ∈ SX
for x ∈ SX
for x ∈ SX
Question 54 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is P(X = 2|Y = 3)?
1.
5
9
2.
1
4
3.
3
7
Question 55 Let the state space of X be SX = {1, 2} and the state space of Y be
SY = {1, 2, 3, 4}. Let the joint pmf of X and Y be defined as
x+y
for x ∈ SX , y ∈ SY
32
f (x, y) =
0
otherwise
What is P(Y ≥ 3|X = 1)?
1.
9
14
2.
10
14
3.
3
7
12
Question 56 Suppose X is a random variable. Define Y = a + bX where a ∈ R and
b > 0. What is ρ(X, Y )?
1. ρ(X, Y ) = 1
2. ρ(X, Y ) = 0
3. ρ(X, Y ) = −1
Question 57 Suppose X is a random variable. Define Y = a + bX where a ∈ R and
b < 0. What is ρ(X, Y )?
1. ρ(X, Y ) = 1
2. ρ(X, Y ) = 0
3. ρ(X, Y ) = −1
Question 58 Suppose X1 , X2 and X3 are independent Poisson random variables with
parameters λ1 = 1, λ2 = 2 and λ3 = 3 respectively. Let X = X1 + X2 and
Y = X2 + X3 . What is Cov(X, Y)?
1. 2
2. 0
3. 6
Question 59 Suppose the number X of books sold in a day in the University bookshop is a random variable with expected value 16. Which upper bound does Markov’s
inequality provide for the probability that sales exceed 20 books?
1.
16
20
2.
16
21
3.
16
19
Question 60 Suppose the number X of books sold in a day in the University bookshop
is a random variable with expected value 16 and variance 9. Which lower bound does
Chebychev’s inequality provide for the probability that sales lie between 11 and 21
books inclusively?
1.
1
2
2.
2
3
3.
3
4
Question 61PLet X1 , . . . , Xn be iid Poisson random variables with parameter 10. Den
fine Sn =
i=1 Xi . According to the Central Limit Theorem which is the Normal
distribution approximating the distribution of Sn ?
13
1. N (0, 1)
2. N (10n, 10n)
3. N (10, 10
n)
Question 62 Let x1 , . . . , xP
n be observations of n iid Bernoulli trials with success
n
probability p. Let x̄ = n1 i=1 xi . Which of the following are likelihood functions
for p?
1. pnx̄ (1 − p)n(1−x̄)
n
2.
pnx̄ (1 − p)n(1−x̄)
nx̄
3. px̄ (1 − p)n−x̄
Question 63 Let x1 , . . . P
, xn be observations of n iid Bernoulli trials with success
n
probability p. Let x̄ = n1 i=1 xi . What is the MLE for p?
1. p̂ = x̄
2. p̂ = nx̄
3. p̂ = nx̄(1 − x̄)
Question 64 Let x1 , . . . , xn be an iid random sample of the distribution with density
function
x
1
x exp(− )
for 0 < x < ∞.
fθ (x) =
θ2
θ
Let x̄ be the sample average. What is the MLE for θ?
1. θ̂ = x̄
2. θ̂ = nx̄
3. θ̂ =
x̄
2
Question 65 Let x1 , . . . , xn be an iid random sample of the distribution with density
function
1 2
x
x exp(− )
for 0 < x < ∞.
fθ (x) =
2θ3
θ
Let x̄ be the sample average. What is the MLE for θ?
1. θ̂ = nx̄
2. θ̂ = x̄
3. θ̂ =
x̄
3
14
Question 66 Let x1 , . . . , xn be an iid random sample of the distribution with density
function
fθ (x) = θxθ−1
for 0 < x < 1.
Let x̄ be the sample average. What is the MLE for θ?
1. θ̂ = x̄
2. θ̂ = − log(Qnn
i=1
3. θ̂ = −n
Pn
i=1
xi )
log(xi )
Question 67 Let x1 , . . . , xn be an iid random sample of the distribution with density
function
θ |x|
fθ (x) =
(1 − θ)1−|x|
for − 1 < x < 1.
2
What is the MLE for θ?
1. θ̂ =
2. θ̂ =
3. θ̂ =
Pn
i=1 |xi |
P
2n− n
i=1 |xi |
Pn
i=1
|xi |
n
Pn
i=1
|xi |
2n
Question 68 Suppose the following is a random sample of a Poisson distribution with
parameter λ.
1
2
1
2
1
0
4
1
0
0
1
1
What is the MLE for λ?
1.
7
6
2.
1
2
3. 2.6
Question 69 Suppose the following is a random sample of a Poisson distribution with
parameter λ.
4
6
9
10
7
3
What is the MLE for λ?
1. 6.2
2. 6.5
3. 7.2
15
6
11
10
6
Question 70 Let Y ∼ B(100, p). To test H0 : p = 0.08 against H1 : p < 0.08 we
reject H0 and accept H1 if and only if Y ≤ 6. What is the significance level of this
test?
100
1.
0.086 0.9294
6
P6
100
2.
0.08k 0.92100−k
k=1
k
P6
100
3.
0.08k 0.92100−k
k=0
k
Question 71 Let p be the probability that a student in a group of 25 obtains a first
class result in his exam. Since p = 0.4 the students decide to take some revision
classes. Then they each take a trial exam. In the trial 13 students obtain a first class
result. You would like to test whether the revision classes have improved the student’s
results. What is the significance level of the appropriate test?
P25
25
1.
0.4k 0.625−k
k=13
k
P13
25
0.4k 0.625−k
2.
k=0
k
P20
25
3.
0.4k 0.625−k
k=6
k
Question 72 Let Y be B(192, p) distributed. We reject H0 : p = 0.75 and accept
H1 : p > 0.75 if and only if Y ≥ 152. Using the normal approximation provided by
the Central Limit Theorem what is the significance level of the test?
1. 0.0918
2. 0.0505
3. 0.0250
Question 73 Let p equal the proportion of drivers who use a mobile phone without
hands-free kit while driving. It is claimed that p = 0.14. Now a fine is introduced and
you would like to test whether this is helpful in reducing the above proportion. Define
the appropriate null and alternative hypothesis.
1. H0 : p = 0.14 against H1 : p 6= 0.14
2. H0 : p = 0.14 against H1 : p > 0.14
3. H0 : p = 0.14 against H1 : p < 0.14
Question 74 Let Y be B(100, p) distributed. Using the normal approximation provided by the Central Limit theorem determine the critical value for H0 : p = 0.5
against H1 : p < 0.5 with significance level α = 0.05.
16
1. 41
2. 9
3. 58
Question 75 It was claimed that the proportion of young British who binge drink is
p = 0.25. You would like to test H0 : p = 0.25 against H1 : p > 0.25. You take a
random sample of size n = 432 and out of these 120 admit to binge drinking. Using
the normal approximation provided by the Central Limit theorem would you reject H0
at significance level α = 0.1?
1. Yes
2. No
Question 76 It was claimed that the proportion of young British who binge drink is
p = 0.25. You would like to test H0 : p = 0.25 against H1 : p > 0.25. You take a
random sample of size n = 432 and out of these 120 admit to binge drinking. Using
the normal approximation provided by the Central Limit theorem would you reject H0
at significance level α = 0.05?
1. Yes
2. No
Question 77 It was claimed that the proportion of young British who binge drink is
p = 0.25. You would like to test H0 : p = 0.25 against H1 : p > 0.25. You take a
random sample of size n = 432 and out of these 120 admit to binge drinking. Using
the normal approximation provided by the Central Limit theorem what is the p-value
for this test?
1. 0.0918
2. 0.0808
3. 0.0418
Question 78 Assume the IQ of experts in Scientific Computing is approximately N (µ, 100).
To test H0 : µ = 110 against H1 : µ > 110 we test n = 16 experts and observe a
sample mean of x̄ = 113.5. Do we reject H0 at a significance level of α = 0.05?
1. Yes
2. No
Question 79 Assume the IQ of experts in Scientific Computing is approximately N (µ, 100).
To test H0 : µ = 110 against H1 : µ > 110 we test n = 16 experts and observe a
sample mean of x̄ = 113.5. Do we reject H0 at a significance level of α = 0.1?
1. Yes
17
2. No
Question 80 Assume the IQ of experts in Scientific Computing is approximately N (µ, 100).
To test H0 : µ = 110 against H1 : µ > 110 we test n = 16 experts and observe a
sample mean of x̄ = 113.5. What is the p-value of this test?
1. 0.3632
2. 0.0808
3. 0.9192
Question 81 Assume the IQ of experts in Scientific Computing is approximately N (µ, 100).
To test H0 : µ = 110 against H1 : µ 6= 110 we test n = 16 experts and observe a
sample mean of x̄ = 113.5. Do we reject H0 at a significance level of α = 0.05?
1. Yes
2. No
Question 82 Assume the IQ of experts in Scientific Computing is approximately N (µ, 100).
To test H0 : µ = 110 against H1 : µ 6= 110 we test n = 16 experts and observe a
sample mean of x̄ = 113.5. Do we reject H0 at a significance level of α = 0.1?
1. Yes
2. No
Question 83 Assume the IQ of experts in Scientific Computing is approximately N (µ, 100).
To test H0 : µ = 110 against H1 : µ 6= 110 we test n = 16 experts and observe a
sample mean of x̄ = 113.5. What is the p-value of this test?
1. 0.0808
2. 0.1616
3. 0.9192
Question 84 Let X be a random variable taking values 1 and 2. Let θ be a parameter
that takes values in {1, 2, 3}. Suppose our prior for θ is given by
P(θ = 1) = 0.2
P(θ = 2) = 0.4
P(θ = 3) = 0.4.
The pmf of X given parameter θ is given by
Pθ (X = 1) = 0.25
Pθ (X = 2) = 0.75
Pθ (X = 1) = 0.5
Pθ (X = 2) = 0.5
for θ = 2
for θ = 1
Pθ (X = 1) = 0.8
Pθ (X = 2) = 0.2
for θ = 3
Given a single observation x = 1 for X what is the posterior probability for θ = 1?
18
1.
40
57
2.
5
57
3.
1
20
Question 85 Let X be a random variable taking values 1 and 2. Let θ be a parameter
that takes values in {1, 2, 3}. Suppose our prior for θ is given by
P(θ = 1) = 0.2
P(θ = 2) = 0.4
P(θ = 3) = 0.4.
The pmf of X given parameter θ is given by
Pθ (X = 1) = 0.25
Pθ (X = 2) = 0.75
for θ = 1
Pθ (X = 1) = 0.5
Pθ (X = 2) = 0.5
for θ = 2
Pθ (X = 1) = 0.8
Pθ (X = 2) = 0.2
for θ = 3
Given a single observation x = 2 for X what is the posterior probability for θ = 2?
1.
1
10
2.
1
25
3.
20
43
19