Instructions
As usual, please upload one file on Blackboard including your answers/solutions to problems. For the R problems, include your input and output. A random subset of problems will be graded but you need to do all the problems (unless you trust your luck (probably you shouldn’t)).
R functions you may need
We will use R for problems regarding binomial and Poisson distributions. In addition to R commands we have learnt previously, the following may help. In order to get help for any of the below functions, you can use the help function in R. For example, with typing help(dbinom), you can get help about the dbinom function.
Binomial distribution
Suppose X is a binomial random variable with parameters n and p. (Think of X as the number of successes you will have in n experiments, where the success probability is p for each experiment, independently of each other.) For 0 ≤ j ≤ n, P (X = j) =pj(1 p)n−j.
• dbinom(x, size, prob) (probability density function) gives the probability P (X = xi) for all xi in the vector x.
• pbinom(x, size, prob) (cumulative distribution function) gives the probability P (X ≤ xi) for all xi in the vector x.
• qbinom(q, size, prob) (quantile function) gives the smallest integer mi for which P (X ≤ mi) ≥ qi for all qi in the vector q.
Here the parameters are:
– x is a vector of nonnegative integers (can be a single number).
– q is a vector of probabilities.
– size (n) is the number of trials (experiments).
– prob (p) is the probability of success of each trial.
Example 1. Suppose we are tossing a coin 5 times and counting the number of heads, which is given by the binomial random variable with parameters 5 (size) and 0.5 (success rate p).
1. If we type dbinom(0:5, 5, 0.5), we see
> dbinom(0:5, 5, 0.5)
[1] 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125
which shows
P (X = 0) = 0.03125
P (X = 1) = 0.15625
P (X = 2) = 0.31250
P (X = 3) = 0.31250
P (X = 4) = 0.15625
P (X = 5) = 0.03125
Recall that 0:5 gives you the vector (0,1,2,3,4,5).
2. If we type pbinom(0:5, 5, 0.5), we see
> pbinom(0:5, 5, 0.5)
[1] 0.03125 0.18750 0.50000 0.81250 0.96875 1.00000
These numbers show
P (X ≤ 0) = 0.03125
P (X ≤ 1) = 0.18750
P (X ≤ 2) = 0.50000
P (X ≤ 3) = 0.81250
P (X ≤ 4) = 0.96875
P (X ≤ 5) = 1
3. If we type qbinom(c(0.1,0.7), 5, 0.5), we see
> qbinom(c(0.1,0.7), 5, 0.5)
[1] 1 3
This means
P (X ≤ 0) < 0.1 and P (X ≤ 1) ≥ 0.1
P (X ≤ 2) < 0.7 and P (X ≤ 3) ≥ 0.7
(The last line says that getting 2 or fewer heads has a chance strictly smaller than 0.7 but getting 3 or fewer heads has chance at least 0.7)
3. If we type qbinom(c(0.1,0.7), 5, 0.5), we see
> qbinom(c(0.1,0.7), 5, 0.5)
[1] 1 3
This means
P (X ≤ 0) < 0.1 and P (X ≤ 1) ≥ 0.1
P (X ≤ 2) < 0.7 and P (X ≤ 3) ≥ 0.7
(The last line says that getting 2 or fewer heads has a chance strictly smaller than 0.7 but getting 3 or fewer heads has chance at least 0.7)
Poisson distribution
Poisson distribution has one parameter, usually denoted as µ or λ. If Y is a Poisson random variable with parameter µ, then µ is the mean value of Y and for k = 0, 1, 2, . . .
−µ µkP (Y = k) = ek!
Density, cumulative distribution, and quantile functions are given as
•dpois(x, lambda, log = FALSE) (probability density function) gives the probability P (Y = xi) for all
xi in the vector x.
•ppois(q, lambda, lower.tail = TRUE, log.p = FALSE) (cumulative distribution function) gives the probability P (Y ≤ qi) for all qi in the vector q.
•qpois(p, lambda, lower.tail = TRUE, log.p = FALSE) (quantile function) gives the smallest integer mi for which P (Y ≤ mi) ≥ pi for all pi in the vector p. (This is when lower.tail= TRUE. If lower.tail= FALSE, then it gives the largest integer mi for which P (Y ≥ mi) ≥ pi for all pi in the vector p)
These are the analogs of dbinom, pbinom, and qbinom functions discussed above. The parameters are
–x: vector of (non-negative integer) quantiles.
–q: vector of quantiles.
–p: vector of probabilities.
–lambda (our mu=µ): vector of (non-negative) means.
–log, log.p: logical; if TRUE, probabilities p are given as log(p). [Don’t worry about this one for this lab]
–lower.tail: logical; if TRUE (default), probabilities are P (Y ≤ x), otherwise, P (X > x).
Note that the default values for lower.tail and log/log.p are TRUE and FALSE, respectively. So if you don’t need to change the default values, you don’t need to include them in your functions.
Example 2. Suppose X is a Poisson distribution with parameter (mean) 12.
1. To find P (X = 16), you can type
> dpois(16, lambda=12) (or dpois(16, 12)) [1] 0.05429334
2. To find P (X = 11), P (X = 12), P (X = 13) you can type
> dpois(11:13, lambda=12) (or dpois(11:13, 12)) [1] 0.1143679 0.1143679 0.1055704
3. To find P (X = 11) + P (X = 12) + P (X = 13) you can type
ppois(13, lambda=12) - ppois(10, lambda=12) [1] 0.3343062
or
> x= dpois(11:13, lambda=12)
> sum(x)
[1] 0.3343062
Remark. Suppose you have a binomial distributi with parameters n and p. If n is large and p is small, you can use the Poisson distribution with mean np to approximate the binomial distribution.
Problems
Part 1: Do the problems by hand
1. One card is randomly drawn from a standard deck of 52 cards. What is the probability that it is an ace or a spade?
2. A deck of cards is shuffled. True or false, and explain briefly:
(a) The chance that the top card is the jack of clubs equals 1/52.
(b) The chance that the bottom card is the jack of diamonds equals 1/52.
(c) The chance that the top card is the jack of clubs or the bottom card is the jack of diamonds equals 2/52.
(d) The chance that the top card is the jack of clubs or the bottom card is the jack of clubs equals 2/52.
(e) The chance that the top card is the jack of clubs and the bottom card is the jack of diamonds equals 1/52 × 1/52.
(f) The chance that the top card is the jack of clubs and the bottom card is the jack of clubs equals 1/52 × 1/52.
3. Suppose 10% of a population is affected by a disease. And suppose we have a test for this disease that is accurate 90% of the time. If an individual is tested positive, what is the probability that this individual is actually carrying this disease?
4. A pair of fair dice is tossed. Find the probability of getting at most a total of 5.
5. Suppose it is known that in a certain population, 10% of the people are color-blind. If a random sample of 12 people is drawn, find the probability that at least 2 of them are color-blind.
6. The ER of a hospital receives 3 patients in an hour on average. What is the probability that there are 5 visits to the ER in a given two-hour period?
Two fun problems (not to be graded).
(I) Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ”Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
(II) A family has two children. At least one child is a boy born on a Tuesday. What is the probability that the other one is also a boy?
CS 340 Milestone One Guidelines and Rubric Overview: For this assignment, you will implement the fundamental operations of create, read, update,
Retail Transaction Programming Project Project Requirements: Develop a program to emulate a purchase transaction at a retail store. This
7COM1028 Secure Systems Programming Referral Coursework: Secure
Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip
CS 340 Final Project Guidelines and Rubric Overview The final project will encompass developing a web service using a software stack and impleme
