In: Statistics and Probability
(20 pts) Inquiries arrive at an information center according to a Poisson process of rate 2 inquiries per second. It takes a server an exponential amount of time with mean ½ second to answer each query.
(a) (5 pts) If there are two servers. What is the probability that an arriving customer must wait to be served?
(b) (5 pts) Find the mean number of customers in the system and the mean time spent in the system.
(c) (10 pts) If the arrival rate increases to 10 inquiries per second and the number of servers is increased to 6, what is the resulting probability that an arriving customer finds all servers busy? What is the mean total delay for each inquiry? What is the percentage of all queries with waiting time less than 1 second?
In: Statistics and Probability
Suppose I work in a factory which does not have the best safety record. On any given day (independent of all other days) there is a 5% chance that there will be a workplace injury. Let X be the number of days until we have a workplace injury.
What is the distribution of X? Give the name of the distribution and the appropriate parameters.
What is the mean and variance of this distribution? Give your answer as a number,
but include the formulas (or logic) used.
What is the probability that there will be an injury within the next 5 workdays?
i.e. what is
P(X≤5)?
2
Let’s say a few employees have a rather inappropriate betting pool on which day
the next injury will occur. If I placed my wager on day 4, what is the probability
that I win?
In: Statistics and Probability
Based on a poll, 80% of airline passengers slept or rested
during a long flight. Let the random variable x represents the
number of airline passengers among the selected six who slept or
rested.
A. Construct a table describing the probability distribution and
the R Syntax. (2 pts.)
B. Find the probability that at least 2 of the 6 passengers who
slept or rested. (2 pts.)
C. Find the mean and standard deviation for the numbers of
passengers who slept or rested. (2 pts.)
D. Determine whether 4 is a significantly high number of passengers
who slept or rested in a group of 6. Explain. (3 pts.)
Use R:
x P(x)
0
1
2
3
4
5
6
In: Statistics and Probability
USE R-studio TO WRITE THE CODES!
# 2. More Coin Tosses
Experiment: A coin toss has outcomes {H, T}, with P(H) = .6.
We do independent tosses of the coin until we get a head.
Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes.
Define a random variable "tosses_till_heads" that counts the
number of tosses until we get a heads.
```{r}
```
Use the replicate function, to run 100000 simulations of this
random variable.
```{r}
```
Use these simulations to estimate the probability of getting a head
after 15 tosses. Compare this with the theoretical value computed
in the lectures.
```{r}
```
Compute the probability of getting a head after 50 tosses. What do
you notice?
```{r}
In: Computer Science
The number of eggs that a female house fly lays during her lifetime is normally distributed with mean 840 and standard deviation 116. Random samples of size 82 are drawn from this population, and the mean of each sample is determined. What is the probability that the mean number of eggs laid would differ from 840 by less than 30? Round your answer to four decimal places.
If samples of size 39 are taken from a bimodal population, the
distribution of sample means will be approximately normal. How can
I be so sure of this?
A. The Law of Large Numbers says so
B. The Central Limit Theorem says so
C. The data is normal because the problem says
so
D. It is a basic property of probability
In: Math
USE R TO WRITE THE CODES!
# 2. More Coin Tosses
Experiment: A coin toss has outcomes {H, T}, with P(H) = .6.
We do independent tosses of the coin until we get a head.
Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes.
Define a random variable "tosses_till_heads" that counts the
number of tosses until we get a heads.
```{r}
```
Use the replicate function, to run 100000 simulations of this
random variable.
```{r}
```
Use these simulations to estimate the probability of getting a head
after 15 tosses. Compare this with the theoretical value computed
in the lectures.
```{r}
```
Compute the probability of getting a head after 50 tosses. What do
you notice?
```{r}
In: Computer Science
Harry Potter books have become popular with children and adults alike. A recent survey conducted in London revealed that 80% high school students have read the first Harry Potter book. A random sample of 7 London high school students is taken, and the number of students who have read the first Harry Potter book is recorded.
a) Define the random variable of interest and give its distribution, including the values of all the parameters.
X = (Click to select)the sample meanSample size of London high school students who participated in the surveyProbability that a London high school student has read the first Harry Potter bookNumber of London high school students who have read the first Harry Potter book
X ~ (Click to select)BinomialNormalContinuousUniform(n=, p=)
Round answers for parts (b) and (c) to two decimal places. Round answers for parts (d) through (h) to four decimal places.
b) What is the expected number of randomly selected students who have read the first Harry Potter book?
c) What is the variance of number of students who have read the first Harry Potter book?
d) What is the probability that exactly two of the randomly selected students has read the first Harry Potter book?
e) What is the probability that at least two of the randomly selected students have read the first Harry Potter book?
f) What is the probability that no more than five of the randomly selected students have read the first Harry Potter book?
g) What is the probability that between two and seven (inclusive) of the randomly selected students have read the first Harry Potter book?
h) What is the probability that more than seven of the randomly selected students have read the first Harry Potter book?
In: Statistics and Probability
The following table gives the projected number of workers in various categories in Springfield in 2020.
| White | Black | Asian | Latino | |
|---|---|---|---|---|
| Male | 49,813 | 6,089 | 2,786 | 10,042 |
| Female | 36,021 | 6,673 | 2,088 | 5,662 |
| Total | 85,834 | 12,762 | 4,874 | 15,704 |
What is the total number of workers represented in the table?
workers
(a)
Use the table to find the following probabilities.
One of the represented workers is chosen at random. Find the probability that the person is Latino. (Round your answer to five decimal places.)
Pr(Latino) =
One of the represented workers is chosen at random. Find the probability that the person is female. (Round your answer to five decimal places.)
Pr(female) =
One of the represented workers is chosen at random. Find the probability that the person is Latino and female. (Round your answer to five decimal places.)
Pr(Latino and female) =
One of the represented workers is chosen at random. Find the probability that the person is Latino or female. (Round your answer to three decimal places.)
Pr(Latino or female) =
(b)
Use the table to find the following probabilities.
One of the represented workers is chosen at random. Find the probability that the person is male. (Round your answer to five decimal places.)
Pr(male) =
One of the represented workers is chosen at random. Find the probability that the person is black. (Round your answer to five decimal places.)
Pr(black) =
One of the represented workers is chosen at random. Find the probability that the person is male and black. (Round your answer to five decimal places.)
Pr(male and black) =
One of the represented workers is chosen at random. Find the probability that the person is male or black. (Round your answer to three decimal places.)
Pr(male or black) =
(c)
Use the table to find the following probabilities.
One of the represented workers is chosen at random. Find the probability that the person is Asian. (Round your answer to five decimal places.)
Pr(Asian) =
One of the represented workers is chosen at random. Find the probability that the person is white. (Round your answer to five decimal places.)
Pr(white) =
One of the represented workers is chosen at random. Find the probability that the person is Asian and white. (Round your answer to five decimal places.)
Pr(Asian and white) =
One of the represented workers is chosen at random. Find the probability that the person is Asian or white. (Round your answer to three decimal places.)
Pr(Asian or white) =
In: Statistics and Probability
A roommate matching service surveys potential tenants for an apartment complex. They run the survey data through a software program that uses an algorithm to identify whether 2 people are a match. The program follows a binomial distribution and finds a positive match 30% of the time. It is able to compare 100 sets of potential roommates per day
a. What is the probability that the software will find exactly 40 matches in any given day?
b. What is the probability that there will be between 24 and 32 matches in any given day?
c. What is the probability that there will be at least 35 matches
d. Is it appropriate to use the normal distribution to approximate probabilities from the binomial distribution of the number of potential roommate matches? How do you know?
e. Use the binomial distribution to find the probability that there will be at most 28 matches found.
f. Approximate the probability from Part (e) above using the normal distribution, if it is appropriate to do so.
In: Statistics and Probability