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# Binomial

Binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each which yields success with probability p. A success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. The binomial distribution is often used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution. When N is much larger than n, the binomial distribution is a good approximation, and widely used.

Typically the mode of the binomial distribution is equal to [(n + 1)p] where [] is the floor function. However, (n+1)p is an integer and p is either 0 now 1, then the distribution has two mode, (n+1)p and (n+1)p-1. Where p is equal to 0 or 1, the mode will be 0 and n correspondingly. There is no single formula to find the median for a binomial distribution, and it may even be non-unique.

If n is large enough, then the skew of the distribution is not too great. In this case, a reasonable approximation to B(n, p) is given by the normal distribution. The baic approximation can be improved in a simple way by using a suitable continuity correction. The basic approximation generally improves as n increases and is better when p is not near to 0 to 1.

The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with the parameter can be used as an approximation to B(n,p) of the binomial distribution if n is sufficiently large and p is sufficiently small.

### Binomial Distribution of Telephone Bill Outcome

Based upon past experience, 1% of the telephone bills mailed to households are incorrect. A sample of 20 bills is selected. p= probability of success= 0.01 n= number of trails= 20 q= probability of failure = 0.99 a. What is the

### Probability of Binomial Random Variable

Based upon past experience, 1% of the telephone bills mailed to households are incorrect. A sample of 20 bills is selected. a. Is this a valid Binomial experiment? b. What is the mean of the probability distribution? c. What is the standard deviation of the probability distribution? d. What is the probability that 10 of the

### Need help ASAP on Integer/Binomial Programming Homework!!

Solve the linear programming models using either lp_solve (recommended, see linear programming tutorial) or excel solver (Google for details). Include mathematical models. 1. A company has 4 projects to be assigned among 3 employees. Find the assignment that minimizes the total time spent by the employees: Peter Mary

### Binonomial Distribution

Can you please explain to me how to do this; I'm so lost. "As a quality engineer working on the Rangdo production line, you have determined that the probability of producing a defective part is 6%. You want to develop a plan to monitor the quality of this line. Your idea is to take a sample of 20 parts every hour and determ

### Variance and the Standard Deviation

In San Francisco, 30% of workers take public transportation daily. a) In a sample of 10 workers, what is the probability that exactly three workers take the public transport daily? [Hint: It is a Binomial Distribution Problem.] f(3) = b) In a sample of 10 workers, what is the probability that at least three w

### Binomial and Hypergeometric Random Variables

4.188: Given that x is a hypergeometric random variable, compute p(x) for each of the following case: a) N = 8, n = 5, r = 3, x = 2 b) N = 6, n = 2, r = 2, x = 2 c) N = 5, n = 4, r = 4, x = 3 See the rest of the problems attached.

### Using a Binomial Table: Independent Tosses

A fair coin is given n independent tosses and X denotes the number of heads. Use this model to make the following calculations. Use Binomial Table or a TI or Excel (a) If n = 10, P(4<=X<=6) =__________ . (b)If n = 10, P(4<X<7)=________ . (c)If n = 25, P(10<X<=15)=_________ . (d) If n = 25, P(X<12)=______ . (e) If n = 25,

### Binomial Computation Problems

Consider a binomial experiment with n=10 and p=.10 a. Compute f(0) b. compute f(2) c. compute P(x less than or equal to 2) d.compute P(x greater than or equal to 1) e. compute E(x) f. Compute Var(x) and o

### Random Binomial Variable

Let Y be a binomial random variable with parameter p. Find the sample size necessary to estimate p to within 0.05 with probability 0.90 in the following situations: (a) If p is thought to be approximately 0.9. (b) If no information is known about p.

### Binomial Probability and Standard Deviation

A random variable X is binomial with n=15, and p=.2. A) What is the probability that X =4 B) what is the probability that X is at least 2 C) What is the expected value of X? what is the standard deviation of X?

### Normal approximation to binomial, central limit theorem- gambling

1. (A Gambling Example). It costs one dollar to play a certain slot machine in Las Vegas. The machine is set by the house to pay two dollars with probability .45 (and to pay nothing with probability .55). Let Xi be the house's net winnings on the ith play of the machine. Then Sn=?in=1 Xi is the house's winnings after n plays of

### hypergeometric, binomial, and poisson graphs using Excel

Using EXCEL, contruct a a graph showing the entire distribution for: a. Hypergepmetric, where n=4, N=20, and D=5 b. Binomial, where n=15 and p = 0.05 c. Poisson, where np = 1.0

### Binomial Distribution Example

Assume that customers visiting the local ice cream shop are binomially distributed by gender. If ten people come to visit the local ice cream shop tomorrow, determine the following: Binomial distribution 10 n 0.5 p X P(X) cumulative probability 0 0.00098 0.00098 1 0.00977 0.01

### Statistics - Binomial problems: Basic

A TV executive is interested in the popularity of a particular cable TV show. She has been told that a whopping 68% of American households would be interested in tuning in to a new network version of the show. If this is correct, what is the probability that all 6 of the households in her city being monitored by the TV industry

### Statistics - Binomial problems: Mean and Standard Deviation

The workers' union at a certain university is quite strong. About 94% of all workers employed by the university belong to the workers' union. Recently, the workers went on strike, and now a local TV station plans to interview a sample of 10 workers, chosen at random, to get their opinions on the strike. Estimate the number o

### Solving a Binomial Distribution Question

Please help with the following problem. Provide step by step calculations. Looking for the steps to figuring out this problem. Should I use some form of a calculator? Is this a binomial distribution question? A student is taking a 10 question quiz. There are four possible answers to each question. The student decide

### Normal as Approximation to Binomial

In a replication of Mendel's hybridization experiment a student observed 161 peas with yellow pods in 600 offspring peas, rather than the 25% (150 peas) predicted by Mendel. Assuming that Mendel's conclusions are correct estimate the probability of getting at least 161 peas with yellow pods among the 600 offspring peas.

### Mathematics - Binomial Random Variable

Given a binomial random variable X with n=5 and p=0.4, then P(x is at least 2) is 0.317. True or False?

### Binomial Distribution of Rare Events

Can the binomial distribution be used to model the number of rare events that occur over a given time period?

### Robert Smith Case Study

3. Robert Smith of the "Merrill Lynch Pierce Fenner & Smith" families is engaged in full-time fundraising. He has been quite successful at this because he brings a very structured approach to the process: you ask people directly and quickly for money. You don't waste your own time and you don't waste their time. His view is

### Rao-Blackwell Unbiased Theorem

We wish to estimate: var(X.bar) = mp(1 - p)/n. Where S=X1+X2+...+Xn is a random sample from a binomial (m, p) population 1. Find an unbiased estimator of mp(1 - p)/n. Note that since E[X1] = mp, X1/m estimates p well, so we may try X1 (m - X1)/m2 as an estimator of p(1 - p), and a constant multiple of X1(m - X1) as an est

### Binomial Random Variable Functions

1. For a binomial random variable with n=8, pie =0.2, P(3)= 2. For a binomial random variable with n=6, and pie = 0.7, P(5)= 3. A package of seeds states that the probability of germination is 0.92. For a package containing 75 seeds, the mean and variance of the number of seeds germinating are:

### Binomial distribution sample question

Consider a binomial distribution with 15 identical trials, and a probability of success of 0.5 a. Find the probability that x = 2 using the binomial tables b. Use the normal approximation to find the probability that x = 2

### Central and Raw Moments Binomial Expansion

By expanding the binomial in (see equation in attached file) and summing term by term, derive a formula for calculating the kth moment about the mean in terms of the kth and lower order moments about the origin.

### approximation to find binominal with a correction for continuity

Suppose that 13% of the population of the U.S. is left-handed. If a random sample of 180 people from the U.S. is chosen, approximate the probability that fewer than 26 are left-handed. Use the normal approximation to the binomial with a correction for continuity. Round your answer to at least three decimal places. Do not rou

### Maximum likelihood methods

6.1.7 Let the table x 0 1 2 3 4 5 freq 6 10 14 13 6 1 represent a summary of a sample of size 50 from a binomial distribution having n=5. Find the mle of P(X>=3). Please show detail on how to find the mle of a binomial too. T

### Probability - Binomial, Normal

1. The Arizona State Office of Tourism Development compiles information about the scenic attractions visited by out-of-state vacationers. The office reports that 75% of out-of-state vacationers visit the Grand Canyon, 35% visit the Sunset Crater (Meteor Crater), and 20% visit both. What is the probability that an out-of-state va

### Density Function (Poisson, Binomial)

(a) If X ~ Poisson ........, compute Mx(t). (b) Show that if Xn ~ Binomial {see attachment} then Mxm(t) converges to Mx(t) for every t, where X ~ Poisson .......

### Normal approximation to the binomial

Statistics show that 69% of all college seniors have a job prior to graduation. If a random sample of 65 recent college graduates is taken, approximate the probability that at most 44 have a job prior to graduation. Do this in the following two ways: a. Use the normal approximation to the binomial without a correction for conti

### Probability and binomial distributions 20 point value

Acme plumbing supply just received a shipment of 5,000 stainless steel valves, but 50 regular steel valves were also sent. There is no way to tell the difference between the valves. A customer orders 5 stainless steel valves . What is the probability that one or more will be regular steel? What distribution will be used? What ar