STA130A summer 2018 Homework 2
Due at the beginning of class on Monday July 9, 2018
Questions relate to the material in chapter 2.
PLEASE FOLLOW ALL INSTRUCTIONS.
These instructions are for your protection to eliminate the consequences of your homework
getting lost and you getting a 0.
The rst page of your homework is a ‘top sheet’ with only your name, student ID, and
homework number. This top sheet should be a single piece of (regular) paper which is NOT
stapled to the rest of the homework.
All homework pages (except the top sheet) must be stapled (before you come to class).
The second page (which means a new sheet of paper, so not the back side of the top sheet)
should ALSO contain your name, student ID, and homework number. Use the format shown
below.
After I collect homeworks I put all of the top sheets into a folder before passing the home-
works to the grader. If at any point during the quarter there is a homework that you know
you turned in, but it does not show on Canvas, contact me. I will look to see if I have your
top sheet from the homework. If I have your top sheet then I will give you credit for the
homework.
It is your responsibility to make sure every homework assignment you submit has a top
sheet. If you tell me you turned in a homework, but there is no Canvas grade and I don’t
have your top sheet in my folder, then you will get a 0.
On both the rst page (‘top sheet’) and second page write your name and student ID on
the top left and homework number on the top center. For example, for homework 2 if your
name is John Smith, and your student ID is 123456789, then the top of your top sheet AND
rst stapled page should look like this
John Smith Homework 2
123456789
Points lost if you
don’t follow the rule
correct format for name, ID, and homework and number 1
Staple all pages EXCEPT the top sheet 1
If your homework is on paper pulled out of a notebook,
cut o all of the fringes (from the torn horizontal threads
that attached the paper to the notebook). 1
Be kind to the grader.
make sure you write your name clearly (so it is easy to read)
write neatly
2
1. We toss two coins. The sample space is = fhh;ht;th;ttg. Consider the following
two di erent random variables X and Y de ned for this sample space. The random
variable X is de ned as
X =
8
>>>
>>
:
1 if hh
2 if ht
3 if th
4 if tt
(1)
The random variable Y is de ned as the number of heads, i.e.
Y =
8
>>>
>>
:
2 if hh
1 if ht
1 if th
0 if tt
(2)
Any function of one or more random variables is also a random variable. De ne a new
random variable Z as Z = X + Y. Give the de nition of Z in terms of its functional
dependence on the sample space , i.e. de ne Z using the same formats as given above
for X and Y in equations (1) and (2).
2. Let f(y) be the density function and F(y) be the cumulative distribution function (cdf)
for a continuous random variable Y with support ( 1;1). For each of the following
indicate if it either
always true (AT),
never true (NT), or
might be true (MBT), but not necessarily always true.
MBT means there could be a continuous random variableY with support ( 1;1)
where the statement is true, but it would not necessarily hold for all continuous
random variables Y with support ( 1;1). In a few of the questions the answer
would change if the support were di erent.
There is only one question where the answer is MBT, so don’t think too hard
about the ones where your rst answer is NT, but then you try to think that
maybe there could be some unusual density function where it would be true.
If you have a good understanding of what density functions and cdfs are, these question
should be mostly easy.
(a) f(y) = ddyF(y)
(b) F(y) = ddyf(y)
(c) f(y) =yR1F(u)du(d) F(y) =yR1f(u)du
(e) f(y) > 0 for all y3
(f) f(y) 1 for all y
(b) Find the probability density function (pdf).
0 x 1). Figure out what
the other two ways are.
(b) Once you have your list of the three ways to have W = 1, you now have to nd
the probability for each one. Use the fact that Y1 and Y2 are independent and
that each has a Poisson distribution with = 1.
12. Suppose a person during the day, on average, gets 5 text messages every 3 minutes.
Assume this can be modeled with a Poisson process.
(a) At any time of the day, what is the probability that they will receive no text
messages in the next minute?
(b) At any time of the day, what is the probability that they will receive 2 text
messages in the next minute.
13. Phone calls are received at a call center with a rate of 30 calls per hour. Assume this
can be modeled with a Poisson process.
(a) During a 10 minute period what is the probability of receiving 2 calls?
(b) During a 10 minute period what is the probability of receiving at least one call?
14. We have an unlimited supply of light bulbs with probability 1 p that a randomly
chosen light bulb is defective. We screw a bulb into a lamp and switch on the current.
If the bulb works, we stop; otherwise, we try another and continue until a good bulb
is found. What is the probability that at least n bulbs are required?
Hint: The key to recognizing that we can de ne a geometric random variable here is
the fact that there is, what is sometimes called, a \stopping rule". The \experiment"
is de ned as repeating some process until some speci ed event occurs, and then the
process is stopped. The stopping rule here is to stop when a good bulb is found. This
clues us into the fact that the we can use a geometric random variable.
15. A bus is scheduled to arrive at a bus stop at 10:00 a.m., but the actual time of arrival
is a random variable X with a uniform. distribution over the 16-minute interval 9:52 to
10:08.
(a) Suppose a person arrives at the bus stop at 9:52. What is the probability they
will have to wait no more than 10 minutes for the bus to arrive?
(b) Suppose a person arrives at the bus stop at 9:55 and the bus has not arrived yet.
What is the probability they will have to wait no more than 10 minutes for the
bus to arrive?
16. It is important to know when it is appropriate to use a Binomial or a Geometric distri-
bution to model the outcome of an ‘experiment’. For each of the following situations
decide if
(i) X is a Binomial variable, if so give the pmf putting in the correct values for n
and p.
(ii) X is a Geometric variable, if so give the pmf putting in the correct value p.
(iii) X is not a Binomial or a Geometric variable, if so say why (which one or more of
the 4 assumption given in class does not hold)
(a) We have two coins A and B. Coin A has probability 1=3 of landing heads and coin
B has probability 1=4 of landing heads. We ip both coins together repeatedly
until they both land heads. Assume the two outcomes are independent. De ne
X as the number of ips required.
(b) Toss a die 12 times. De ne X as the number of time the die lands on an odd
number.
(c) A man with 20 keys on a key rings wants to open his o ce door. He doesn’t
know which key it is so he tries them one-by-one at random. Exactly one key will
open the door. De ne X as the total number of keys tried including the one that
opened the door.
i. Assume that any unsuccessful key that is tried is eliminated from further
selection.
ii. Assume that any unsuccessful key that is tried is NOT eliminated, but is
instead put back into the pile, and has the same probability of being selected
on the next attempt.
(d) Roll two dice ten times. De ne X as the number of times the two dice do not
show the same number.
(e) Two six sided die are tossed repeatedly until both die show the same number.
De ne X as the number of tosses.
(f) A bag contains 95 functioning and 5 dead batteries. We randomly sample repeated
from the bag without replacement, test the battery, and stop when we nd a good
one. De ne X as the number of batteries sampled.
(g) A box has four red and six white balls.
i. Sample three balls one-by-one with replacement. De ne X as the number of
red balls.
ii. Sample three balls one-by-one without replacement. De ne X as the number
of red balls.