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PSTAT 120A 1 / 2
FINAL EXAM^1 / 2
Summer 2022 Instructor: Ethan Marzban
NAME: Perm Number:
SECTION (circle one): 3:30 - 4:20pm (Lucas) 5 - 5:50pm (Moya) 8 - 8:50am (Moya)
Instructions:
- You will have 170 minutes to complete this exam.
- You are allowed the use of two 8.5 × 11-inch sheets, front and back, of notes. You are also permitted the use of calculators ; the use of any and all other electronic devices (laptops, cell phones, etc.) is prohibited.
- Unless otherwise specified, simplification is not needed; however, all integrals and infinite sums (unless otherwise specified) must be evaluated. - One exception is that, whenever applicable, answers may be left in terms of Φ, the stan- dard normal c.d.f..
- Good Luck!!!
Honor Code: In signing my name below, I certify that all work appearing on this exam is entirely my own and not copied from any external source. I further certify that I have not received any unauthorized aid while taking this exam.
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Multiple Choice Questions: Question: 1 2 3 Total Points: 1 1 1 3 Score:
Short-Answer Questions: Question: 4 5 6 7 8 9 10 11 12 13 Total Points: 7 9 11 11 10 9 5 11 9 5 87 Score:
1 Multiple Choice Questions
Please fill in the bubble(s) on the exam below corresponding to your answer. You do not need to submit any additional work for these questions.
- For (X, Y) [1pts.] i.i.d. ∼ N (0, 1) and a fixed constant z ∈ [0, 1], which of the following correctly provides P(X + Y ≤ z)?
⃝
ż (^) z
0
ż (^) z−x
0
( 1 ) dx dy
ż (^1)
0
ż (^) z−x
0
( 1 ) dx dy
ż (^) z
0
ż (^) z−y
0
( 1 ) dx dy
ż (^1)
0
ż (^) z−y
0
( 1 ) dx dy ⃝ None of the above
- Given a bivariate random vector (X, Y) with joint probability density func- [1pts.] tion (p.d.f.) given by fX,Y (x, y), which of the following correctly computes E[cos(X + Y)]? (Only one answer choice is correct.) ⃝
ż (^) ∞
−∞
cos(x + y) fX,Y (x, y) dx
żż
R^2
cos(x + y) fX (x) fY (y) dA
⃝
żż
R^2
cos(x + y) fX,Y (x, z − x) dA
⃝
żż
R^2
cos(x + y) fX,Y (x, y) dA ⃝ None of the above
2 Short Answer Questions
Please mark your final answers in the spaces provided below each question. Be sure to show all of your work!
- The archipelago of Gauchonia is a collection of 12 smaller islands and one “Main Island,” on which the capital lies. A ferry exists to help locals commute from island to island. On one particular trip, 10 people board the ferry at the cap- ital on the Main Island, and then request to disembark on a random island, independently of all other passengers. Assume the following: - Nobody disembarks at the Main Island - Nobody can stay on the ferry forever (i.e. everyone disembarks at some island) - The ferry only stops at an island if someone is disembarking at that island. Let X denote the number of islands at which the ferry makes a stop.
(a) What is P(X = 12 )? [2pts.]
(b) Compute E[X]. As a hint: Elevator Problem. [5pts.]
- Let (X, Y) be a bivariate random vector with joint probability density function (p.d.f.) given by
fX,Y (x, y) =
· xy if
y ≤ x ≤ 2, 0 ≤ y ≤ 1
0 otherwise
(a) Show that fX,Y (x, y) is a valid joint p.d.f. Be sure to show ALL of your [4pts.] steps fully; large lapses in logic will result in point penalties.
(b) Set up, but DO NOT EVALUATE a double integral (or set of double inte- [3pts.] grals) corresponding to P(X + Y ≤ 2 ).
(b) Identify the distribution (by name!!) of X^2 (remember that X, the x−coordinate [5pts.] of the point, follows the distribution with p.d.f. given above). Include any/all relevant parameter(s).
(c) Identify the distribution (by name!!) of X^2 + Y^2. Include any/all relevant [3pts.] parameter(s). As a reminder, you can use previously-derived results with- out proof so long as you cite them.
- Let (X, Y) be a bivariate random vector with joint probability density function (p.d.f.) given by
fX,Y (x, y) =
λ ye−y(x+ λ )^ if x ≥ 0, y ≥ 0 0 otherwise
where λ > 0 is a fixed constant.
(a) Find fY (y), the marginal p.d.f. of Y and use this to identify Y as belonging [4pts.] to a known distribution. Be sure to include any/all relevant parameter(s)!
(b) Find fX|Y (x | y), the density of (X | Y = y), and use this to identify [3pts.] (X | Y = y) as belonging to a known distribution. Be sure to include any/all relevant parameter(s)!
(b) Suppose that it is also known that the standard deviation of the time a [3pts.] randomly selected students takes to complete the exam is 5 minutes. Use Chebyshev’s Inequality to bound the probability that a randomly selected student will complete the exam in under 31 minutes. Be sure to clearly state whether this is an upper or lower bound!
(c) Now, suppose 64 students are sampled at random (and with replacement), [4pts.] and the average time to complete the final exam among these 64 students is recorded. Approximately what is the probability that the average com- pletion time was lower than 31 minutes?
- Doctor Strange is travelling through the Multiverse, visiting parallel universes one by one. There is, however, a 25% chance that any one of his trips will cause an Incursion, independently of all of his other trips.
(a) What is the probability that exactly 3 of the first 5 trips Doctor Strange [3pts.] makes result in Incursions?
(b) What is the probability that the 7th^ trip Doctor Strange makes is the first [3pts.] trip that results in an Incursion?
- Suppose (X, Y) is a discrete bivariate random vector with joint probability mass function (p.m.f.) given by
Y
X
Furthermore, define events A, B, C in the following manner (recall that 0 is an even number):
A = {X is even} B = {Y is odd} C = {Y is a multiple of 3}
(a) Compute P(A) [3pts.]
(b) Compute P(A ∪ B) [4pts.]
(c) Are A, B, C mutually independent? Justify your answer mathematically, [4pts.] using the definition of mutual independence.
- The speed of a randomly selected car travelling along I-5 is a random variable X with MGF (Moment-Generating Function) given by
MX (t) = exp
70 t +
t^2
; ∀t ∈ R
Additionally, the posted speed limit is 65mph.
(a) What is the average speed of a randomly selected car travelling along I-5? [2pts.]
- Consider a random variable X with mean μ and variance σ^2. If f is an ap- [5pts.] propriately differentiable function, use a second-order Taylor Series Expansion (i.e. a Taylor Series Expansion taken out to the third term in the sum) to show that E[ f (X)] ≈ f ( μ ) +
σ^2 2 f ′′( μ )
COMMON MGF’s:
Distribution MGF at t
Bin(n, p) ( 1 − p + pet)n, ∀t ∈ R
Geom(p)
pet 1 − ( 1 − p)et^ if t < − ln( 1 − p)
∞ otherwise
NegBin(r, p)
pet 1 − ( 1 − p)et
r if t < − ln( 1 − p)
∞ otherwise Pois( λ ) e λ (e t (^) − 1 ) , ∀t ∈ R
Distribution MGF at t
Exp( λ )
λ λ − t
if t < λ 0 otherwise
Gamma(r, λ )
λ λ − t
r if t < λ
0 otherwise
N ( μ , σ^2 ) exp
μ t + σ^2 2
t^2
; ∀t ∈ R
Unif[a, b]
etb^ − eta t(b − a) if t ̸= 0 1 if t = 0
USEFUL RESULTS FROM MATHEMATICS:
( Please note that these are by no means comprehensive; I expect you to know all of the sums from the Calculus Review Series, as well as common mathematical results. )
n
k= 0
n k
akbn−k^ =
n
k= 0
n k
an−kbk
ex^ + e−x 2 ; sinh(x) :=
ex^ − e−x 2
∞
n= 0
f (n)(a) n! (x − a)n^ = f (a) + (x − a) f ′(a) +
(x − a)^2 f ′′(a) +
(x − a)^3 f ′′′(a) + · · ·
You may use this page for scratch work, if necessary.
You may use this page for scratch work, if necessary.
