Book Homework
1. A doctor knows from experience that the probability a
patient shows side effects on a particular drug is 0.10. Assume
that probability does not change, and that patients
are independent.
(a) How many patients on average should he expect to
give the drug before one of them has a side effect?
(b) What is the probability that the 4th patient he gives
the drug to is the first to have a side effect?
(c) What is the probability that the 2nd patient to have
a side effect occurs between the 4th, 5th, and 6th
patient to receive the drug?
(d) What is the probability that the 3nd patient to have a
side effect occurs exactly on the 6th patient to receive
the drug?
2. At a particular manufacturing plant, the probability that
a created item does not meet specifications is 0.04. The
assembly will stop for recalibration if it detects 2 items
that do not meet specifications. Assume items are independent,
and the probability of not meeting specifications
does not change.
(a) Find the expected number of items produced before
the machine needs recalibration, and the standard
deviation.
(b) What is the probability the machine stops on exactly
the 8th item?
(c) What is the probability the machine stops after the
4th item?
(d) What is the probability the first item that does not
meet specifications is the 4th item?
3. Assume that IQ is normally distributed, with mean 100
and standard deviation 15.
(a) What is the probability that a randomly selected persons
IQ is over 120?
(b) Find the values of Q1, Q2, and Q3 for IQ.
(c) Find the values of lower = Q1 ? 1.5(Q3 ? Q1) and
upper = Q3 + 1.5(Q3 ? Q1) for IQ. Recall that these
were cutoffs for outliers.
(d) Find the probability of an outlier for IQ for a single
person based on your values for (c).
(e) If we randomly selected 10 people, what is the probability
their average IQ is over 105?
4. Assume that the selling price of a particular item (call
it I1) is normally distributed, with mean 200 gold coins,
and standard deviation 50 coins. Assume the selling price
of another item is also normally distributed, with mean
230 gold coins, and standard deviation 10 gold coins (call
it I2). Assume these distributions are independent.
(a) Find the probability the first item sold for more than
the second item.
(b) Find the probability that both items sold for over
400 gold pieces total.
(c) Find the probability that the average of the two items
sold for between 220 and 250 gold pieces.
(d) Find the 30th percentile for the average price of the
two items.
(e) Find the 60th percentile for the total of the two
items.
(f) Find the value of the total price for both where only
10% of the time, both items will sell for more than
that value.
5. The average amount of time until a car accident on a
particular 60 mile stretch of road is 60 minutes. Assume
(unreasonably) that car accidents are independent, and
that two accidents cannot occur at the same time.
(a) What is the probability of a car accident occurring
in the two hours?
(b) What is the probability of a car accident occurring
between 45 and 75 minutes?
(c) What is the variance of the time until a car accident
occurs?
(d) If a car accident has not happened in 3 hours, what is
the probability it will happen in the next two hours?
6. In a particular state, a major flood happens once every 3
years on average. If you can assume floods are independent,
and that two floods cannot happen simultaneously,
find the following:
(a) The probability that a flood happens within the first
5 years.
(b) The probability that a flood happens after the first
2 years.
(c) If there has been no flood for 4 years, what is the
probability there is a flood in the next 2 years?
(d) The expected time until a flood happens, and the
standard deviation.
(e) Find the 50th percentile of the time until a flood.
7. The wait time at a popular fast food restaurant is uniformly
distributed between 1 and 15 minutes on average.
(a) Find the probability that you wait less than 6 minutes.
(b) Find the probability that you wait between 5 and 10
minutes.
(c) What is the expected wait time, and the variance of
wait time?
(d) If you have waited 5 minutes already, what is the
probability you wait an additional 5 minutes?
8. The total amount of tips a waitress receives is uniformly
distributed between $1 and $20 in an hour on average.
1
(a) Find the probability the waitress receives between $5
and $10.
(b) What is the expected tip amount the waitress will
receive, and the standard deviation?
(c) Find the following probabilities: P r(X < µX),
P r(X > µX) where X =the amount of tips a waitress
receives.
(d) Find the following probabilities: P r(X < µX ? ?X),
P r(X > µX + ?X) where X =the amount of tips a
waitress receives.
(e) What is the median amount of tips a waitress will
receive?
R Homework
I. The goal of this problem is to simulate a Geometric random
variable. Assume in a computer game Stardew Valley,
there is a 10% chance it rains on any given day.
(a) Use R to simulate generating days up to and including
the first time it rains. Return based on one (random)
run of your simulation the number of days it
took up to and including the first rainy day.
(b) Repeat (a) 20000 times, so that you should have
20000 values of how many days it took up to and
including the first rainy day. Do not print out
these values!!! Plot a histogram of these values.
(c) What can you say about the distribution of this
particular Geometric random variable based on (b)?
Describe it.
(d) Find the expected value and standard deviation of
your vector in (b).
(e) Find the probability that it took 15 days until the
first rainy day.
(f) Find the probability that if it took more than 10
days, it would take an additional 5 days to observe
the first rainy day.
II. The goal of this problem is to simulate a Negative Binomial
random variable. Consider that 40% of people
wear glasses for near-sightedness.
(a) Use R to simulate selecting random people until 5
are selected wear glasses. Return one the number of
people it took until 5 wore glasses.
(b) Repeat (a) 20000 times, so that you should have
20000 values of how many people were were selected
until 5 wore glasses. Do not print out these values!!!
Plot a histogram of these values.
(c) Find the median of the vector from (b).
(d) Find the mean and standard deviation of your vector
from (b).
(e) Find the probability we had to sample more than 10
people before we found 5 that needed glasses.
(f) Find the probability that if it takes more than 10
people, it will an additional 5 to find 5 that wore
glasses.
III. Consider the function rnorm. This simulates a normal
random variable, for example the following code will generate
1000 normal random variables with mean 5, standard
deviation 6:
X = rnorm(1000, mean = 5, sd = 6)
(a) Generate 10000 values of a normal random variable
with mean 2, standard deviation 1, and call this vector
X. Generate 10000 values of a normal random
variable with mean -2, standard deviation 5, and
call this vector Y . Report back the mean and standard
deviation of both vectors (using the functions
in R).
(b) Create a vector W by adding the two vectors from
(a) together, and find the mean and standard deviation
W (using R).
(c) Find the probability that W is larger than 3.
(d) We know what the answers to (b), (c) should be
exactly. Calculate the error for all the values found
(the error is what R found subtract by what we know
it should be).
2
ScholarMatic: Explanation & Answer
Your ready answer from a verified tutor is just a click away for as little as $14.99
Click Order Now to get 100% Original Answer Customized to your instructions!
