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Name:
Stat 311 Sample Midterm Examination
Part I. Do all problems in the space provided. Clearly define all random variables and other notation that you use, and clearly specify what you are calculating at each step in a calculation. Justify all answers. Clearly mark all answers.
- (8 pts) A coin is flipped three times. Let
A = {first flip is heads} B = {second flip is heads} C = {third flip is heads}
Express each of the following events in terms of A, B, and C.
D = {all three flips are tails} = Ac^ ∩ Bc^ ∩ Cc
E = {exactly one flip is tails} = (Ac^ ∩ B ∩ C) ∪ (A ∩ Bc^ ∩ C) ∪ (A ∩ B ∩ Cc)
- (12 pts) Let P (A) = .6 and P (B) = .4.
(a) If A and B are disjoint, what is P (A ∪ B)?
P (A ∪ B) = 1
(b) If A and B are independent, what is P (A ∪ B)?
P (A ∪ B) = .6 +. 4 −. 6 × .4 =. 76
(c) If B ⊂ A, what is P (A ∪ B)? P (A ∪ B) =. 6
- (12 pts) Suppose a random variable X has a cumulative distribution function (cdf) given by
FX (x) = e
x 1 + ex (a) What is the probability density function (pdf) for X?
fX (x) =
(1 + ex)ex^ − exex (1 + ex)^2 =^
ex (1 + ex)^2
(b) Calculate P { 0 < X < 3 }.
P { 0 < X < 3 } = e
3 1 + e^3
= e
2(1 + e^3 )
(c) Calculate P {− 1 < X < 1 |X > 0 }.
P {− 1 < X < 1 |X > 0 } = P^ {^0 < X <^1 }
P {X > 0 }
e 1+e −^
1 2 1 2
= e^ −^1 1 + e
- (8 pts) Suppose Y has a discrete distribution with pmf given by
y 0 1 2 3 pY (y) 1 / 2 1 / 4 1 / 8 1 / 8
(a) What is R(Y )? R(Y ) = { 0 , 1 , 2 , 3 }
(b) Calculate P (Y > 0 }? P {Y > 0 } =^1 2
(c) Calculate E[(Y − 1)^2 ]?
E[(Y − 1)^2 ] =
2 ×^ 1 +
4 ×^ 0 +
8 ×^ 1 +
8 ×^ 4 =
- Sam has 2 coins. One is fair (equal probability of heads and tails) and the other is a trick coin that comes up heads with probability 14. Sam selects one of the coins at random and repeatedly flips the coin. (a) If the first 4 flips include exactly two heads, what is the probability that the fair coin was selected? A = {fair coin selected}, S 4 = # of heads in first four flips Then P (A) = 12 , P (Ac) = 12 and
P {S 4 = 2|A} =
( 4 2
) 1 24
P {S 4 = 2|Ac} =
( 4 2
) ( 1 4
) 2
so P (A|S 4 = 2) =
P (S 4 = 2|A)P (A)
P (S 4 = 2|A)P (A) + P (S 4 = 2|Ac)P (Ac)
= 1 /^2
4 1 / 24 + 3^2 / 44
=^16
(b) If the first 4 flips are heads, what is the probability that the 5th flip is heads? S 5 =# heads in first five flips
P {S 4 = 4} = P ({S 4 = 4} ∩ A) + P ({S 4 = 4} ∩ Ac) =
P {5th flip heads|S 4 = 4} = P^ {S^5 = 5} P {S 4 = 4}
( (^1) 2
) (^5 ) 2 +
( (^1) 4
) (^5 ) 2 ( (^1) 2
) (^4 ) 2 +
( (^1) 4
) (^4 ) 2 =
(c) If the first n flips are heads, what is the probability that the next flip is heads? (You do not need to simplify the expression.) What is the limit as n → ∞?
P {flip n + 1 is heads|Sn = 4} = P^ {Sn+1^ =^ n^ + 1} P {Sn = n}
=
( (^1) 2
)n+1 1 2 +
( (^1) 4
)n+1 1 ( 2 1 2
)n (^1) 2 +
( (^1) 4
)n (^1) 2 →
Name:
- A bucket contains four balls, numbered 1, 2, 3, and 4. The balls are drawn one at a time. Let X be the sum of the first two numbers drawn and Y the sum of the first three.
(a) Calculate the joint probability mass function for X and Y. Display the answer as a table. R(X) = { 3 ,... , 7 } R(Y ) = { 6 ,... 9 } S = {(a 1 , a 2 , a 3 ) : ai ∈ { 1 , 2 , 3 , 4 }, ai 6 = aj for i 6 = j} #S = 4 × 3 × 2= 24
x y^6 7 8 3 242 242 0 0 4 242 0 242 0 5 242 242 242 242 6 0 242 0 242 7 0 0 242 242
(b) Calculate P {X = 3|Y = 7}.
P {X = 3|Y = 7} =
P {X = 3, Y = 7}
P {Y = 7} =
Name:
- If a deli owner charges c dollars per pound for pickled hering, the amount she can sell during a day has an exponential distribution with parameter 10+c
2
- The cost to make the pickled hering is $3 per pound. Since pickled hering only gets better with age, there is no wastage.
(a) How much should the deli owner charge to maximize her expected profit? X = demand Y = profit
E[Y ] = E[(c − 3)X] = (c − 3)
10 + c^2 ≡^ G(c)
G′(c) = (10 +^ c
(^2) )2000 − ((c − 3)20002c (10 + c^2 )^2 =^
(10 + c^2 )^2 (10 + 6c^ −^ c
c = −6 +^
(b) Suppose the deli owner has room for only 50 pounds of pickled hering and cannot replenish her supply during the day. How much should she charge to maximize her expected profit? Derive the equation that you must solve to find the optimal price, but you do not need to solve it. Will the optimal price be more or less than the price determined in Part 5a? Explain your answer.
Y =
{ X(c − 3) X ≤ 50 50(c − 3) X > 50
and setting λ = 10+ 2000 c^2
E[Y ] =
∫ (^50)
0
(c − 3)xλe−λxdx +
∫ (^) ∞
50
50(c − 3)λe−λxdx
= −(c − 3)xe−λx|^500 +
∫ (^50) 0
(c − 3)e−λxdx
∫ (^) ∞ 50
50(c − 3)λe−λxdx
=
c − 3 λ (1^ −^ e
−λ (^50) )
= G(c)(1 − e−^ 10+c
2 (^2000 50) )
Solve G′(c)(1 − e−^ 10+c
2 (^2000 50) ) + G(c) 10 +^ c
2 40
