SPIPractice Sols
University of Illinois
Stat 400 – Exam 1
Instructor: Albert Yu
Time Limit: 75 minutes
Name:
Solutions
-
Form A
(Stat)
NetID: (e.g. ayu12 )
Section: (circle one) 9:30am 11:00 am
INSTRUCTIONS:
• Keep your eyes on your own paper or straight ahead. Your face must be visible to the proctors.
Hats must be turned backwards.
• Please keep your exam papers covered when you are not writing.
• You are not allowed to give or receive any assistance to/from another person for this exam.
Please do not discuss the contents of this exam in Discord until all conflict exams have been
taken (there will be an announcement).
• Allowable materials: Graphing calculators, writing utensils, blank scratch paper, and a
single 2-sided hand-written cheat sheet 8.5 x 11 inch maximum. (No printed notes allowed).
• The word evaluate means you need to include a numerical answer (either a decimal or a
reasonably simplified fraction).
• Please show your process (work) for full / partial credit.
• Please pace yourself and in general do not spend more than 10 minutes on any single question.
• Algebra must be done by hand, even if you are using a fancy calculator to get the answers. If
insufficient or no work is shown, it is possible for a correct answer to receive no credit.
This exam contains 10 pages and 7 questions. There are (100) points possible.
Good luck! :)
1
Stat 400
Exam 1
Question
Points Score
1
3
2
9
3
18
4
27
5
18
6
12
7
13
Total:
100
The space below may be used as scratch paper
Page 2 of 10
Stat 400
Exam 1
1. (3 points) This is not a question. Welcome to Exam 1! Everyone will get 3 points automatically
for writing their name and netID clearly and correctly on the cover page :) (we will not
give points if it is blank, or if a UIN is written instead).
2. In a random experiment, let A and B be events where P (A) = 0.9, P (B) = 0.8, and P (A\B) =
0.75.
(a) (3 points) What is the probability that at least one of these two events will occur?
(b) (6 points) Given that you know event A has occurred, what is the probability that event
B will occur?
a) Solve for P(AUB) P(A) P(B)
8.9
=
3) Solve for P(B/A)
8.8
+
P(A1B)
-
+
=
-
0.75
P(B(A)
=
=
P(A)
Page 3 of 10
0.95
=
0.75
5/6
=
0.833
=
Stat 400
Exam 1
3. Suppose you have a loaded coin with P [T ails] = 0.6.
Parts (a) - (d): Suppose you flip this loaded coin 5 times. Evaluate the probability of the
following events:
(a) (3 points) {H, H, H, H, H}
(b) (3 points) {H, T, H, T, H} (in this exact order )
(c) (4 points) {At least 1 heads}
(d) (4 points) {Exactly two tails}
(e) (2 points) Can we assume the outcomes for each individual toss are pairwise independent?
(f) (2 points) Can we assume the outcomes for each individual toss are mutually independent?
a) P(H) 1 -p (Tails) 1 0.6 0.4
=
=
-
=
P(EH, H, H,
H,
H3) (8.4)5
=
b) P(SH, T, H, T, H3)
(0.4(3(8.6)" 723125
=
1 -p
=
1
=
(because getting an
on
a
second toss
-
0.02304
=
(5 All Tails3)
18.675 28823125
=
0.92224
=
(5)(0.3)"(0.4)" 144625 0.2304
d)P([ Exactly 2Tails3)
Yes,
8.01024
=
=
headh3)
2) P(AtleastI
2)
323125
=
=
=
=
outcome
on
the
firsttoss does
impactthe outcome
not
elaboration not required
↓
2) Yes, (because
the tosses
are
each
pairwise independent,
impactthe
the first toss does not
Page 4 of 10
outcome
on
andan
any
other
outcome on
toss
Stat 400
Exam 1
4. Let
f (x) = c(0.4)x ,
x = 1, 2, 3, ...
Xa f(x) c5
(a) (3 points) Evaluate the value of c that makes this a valid probability distribution.
zc51"
=1
=
=
(2/5) 1
=
c(5) 1
c
->
=
3/2 1.5
=
=
(b) (7 points) Evaluate the expected value of X, E[X]. Advice: If you get stuck on one
part, write a note and make up values for what you are missing. For example, if you could
not solve part (a), make up a value for c.
E(X 3
sk.v" (in) for
Note:
=
peng
2x.f(x) (2,4.(8.4)*
=
=
->
Bar
in answer
from
->
al
cx)
c.()
=
=
(E).(5)
5/3 1.667
=
=
=
(c) (5 points) Evaluate the probability of getting an odd outcome.
x 51,2,3,...3
=
P(8ddOutcome)
Let k be a
->
only
that
transformation of x
k 2x
- >
include odd values
=
1,k 31,3,5....3)
=
so
+
x 50,
=
1,2....3
28c.(0.4) 28c.(0.4(2x* c(210.4)*.10.))
"
=
=
=
c.(0.4).27(0.16)
=
c.(0.4).
=
c()
=
-
(1-0.16)
plug in answer from
5/7
=
Page 5 of 10
0.7143
=
a
Stat 400
Exam 1
(d) (5 points) Evaluate the probability of getting a 1 given that you have an odd outcome.
Solve for Pr /OddOutcome)
P(110ddOutcome)
c(0.4)
P(I)
=
=
PlOddOutsme)
=
PlOddOutcome)
(5k)
plug
in answer
from
a
d.() 1.5.()
=
=
21/25
=
0.84
=
(e) (3 points) How would you find an expression for the moment generating function of X?
Just set up the equation for the summation completely, including the indices. No need to
solve.
SKIP
(f) (4 points) Let Z be a linear transformation of X, such that Z = 8
Evaluate E[Z].
E(z) E(8 4X7 E(8)
-
=
-
=
8
-
=
8
=
-
4X.
E[4x]
4.ETX] prugin
4(5)
4/32 1.333
=
Page 6 of 10
answer
from
b)
Stat 400
Exam 1
5. The number of times your Komkast internet goes out in any given month follows a Poisson
Process with rate = 3.
Suppose there are 12 months in a year and 30 days in a month (and 360 days in a year).
(a) (3 points) What is the standard deviation of the number of times that your internet
will go out in a year?
SKMP
(b) (5 points) Evaluate the probability that your internet will go out either 5 or 6 times
(exactly) in any given month.
(c) (5 points) Assume you get Komkast at the beginning of the year. What is the probability
that the first outage will occur during March (the 3rd month)? Note: This scenario allows
for more than 1 outage in March.
(d) (5 points) Assume you get Komkast at the beginning of the year. What is the probability
that there will be exactly one outage during March (the 3rd month), AND no outages
before that?
Page 7 of 10
Stat 400
Exam 1
6. Let X be a random variable with the following pmf. ( is an unknown constant).
x
0
2
4
f (x)
1 2
(a) (1 point) What is the space of X? (Write your answer as a set).
x 20,2,43
=
(b) (5 points) Find an expression for E[X]. Your answer should be in terms of .
E(x) 2x.f(x)
=
(0)(B) (2)(B) (4)(1 2B)
-
+
=
4
8B
2(2 3B)
25
0
+
=
4
+
6B
-
=
+
-
=
-
(c) (6 points) Find an expression for V ar[X]. Your answer should be in terms of .
VarIx] E(x2]
=
=[x-) (02)(B)
=
(23)(B) (42)(1 2B)
-
+
+
43
0
=
16
(E5X3)3
-
+
Varix) (16 28B)
-
285
-
0 201
+
=
=
-
-
=
16
-
323
-
28B
-
=
=
16
+
3632
-
Page 8 of 10
-
6B)2
(16 3657
+
-
48B)
3332
20B
+
(4
4B(
=
-
95
5)
+
Stat 400
Exam 1
7. Chloe finds a secret stash of doggie treats with the following flavors: 30 beef, 30 chicken, 40
pork. She selects treats at random, licks them, and puts them back in the bag (she can select
the same treat multiple times).
(a) (7 points) Let X represent the number of trials it takes her to get a pork treat for the
10th time. What is the probability that she gets the 10th pork treat on (exactly) the 20th
trial? State the distribution of X and write the equation you need to solve. (No need
to evaluate).
SKp
(b) (6 points) If Chloe selects 7 treats (in total, without replacement), evaluate the probability
that she randomly selects exactly 2 beef and at most 1 chicken treat (in any order).
Page 9 of 10
Stat 400
Exam 1
This page may be used as scratch paper or additional space.
Note: If we need to grade something on this page, do not detach.
Page 10 of 10