Statistical Methods
Bernoulli and Binomial Distribution
Q.1
Identify whether the following experiments form a sequence of Bernoulli Distribution:
(a) Suppose that a student takes a multiple choice test. The test has 10 questions, each of which has 4 possible answers (only one correct). If the student blindly guesses the answer to each question, do the questions form a sequence of Bernoulli trials? If so, identify the trial outcomes and the parameter p. Also calculate mean and variance.
(b) A person wants to model the waiting time (rounded in minutes) before next class, in average he waits 55 minutes. Can he use Bernoulli sequence to model this? If yes, identify the outcomes, parameter, mean, and variance.
(c) Candidate A is running for office in a certain district. Twenty persons are selected at random from the population of registered voters and asked if they prefer candidate A. Do the responses form a sequence of Bernoulli trials? If so identify the trial outcomes and the meaning of the parameter p. Also, calculate mean and variance.
(d) An American roulette wheel has 38 slots; 18 are red, 18 are black, and 2 are green. A gambler plays roulette 15 times, betting on red each time. Do the outcomes form a sequence of Bernoulli trials? If so, identify the trial outcomes, the parameter p, mean, and variance.
(e) A drug has success rate of 95%. This drug is given to one patient, is this Bernoulli trial? If yes identify the parameter and calculate mean and variance.
(f) Mr. A tosses a coin until a head appears. Is this form a sequence of Bernoulli trial? If yes identify the parameter and calculate mean and variance.
Q.2
What three conditions must be satisfied for Binomial distribution?
Q.3.
Based on the conditioned mentioned in Q.2, which of the above problems from Q.1 (a-f) can also be modeled by using Binomial distribution. State reasons. For those which can be modeled by Binomial distribution define random variable X, outline the parameters in standard form (write in X ~ Bin(n,p) format), calculate mean and variance.
Q.4.
Identify which of the followings are binomial distribution (also give the reason why)
I. Rolling a die until a 6 appears
II. Asking 20 people how old they are
III. Drawing 5 cards from a deck for a poker hand (In poker it is without replacement case)
IV. Measuring the tide high or low 20 times.
V. Two of the Hilda's eight friends are vegetarians. Hilda decides to ensure that the eight sandwiches she takes to the match will include at least two suitable for vegetarians. If, having selected eight sandwiches at random, she finds they include fewer than two suitable for vegetarians she will replace one, or if necessary two, of the sandwiches unsuitable for vegetarians with the appropriate number of sandwiches suitable for vegetarians. Interest is in number of sandwiches suitable for vegetarians
Q.5
Jeremy sells a magazine which is produced in order to raise money for homeless people. The probability of making a sale is, independently, 0.09 for each person he approaches. Given that he approaches 40 people, find the probability that he will make:
(a) 2 or fewer sales;
(b) Exactly 4 sales;
(c) ) More than 5 sales.
Q.6
Eight friends take a picnic to a cricket match. As her contribution to the picnic, Hilda buys eight sandwiches at a supermarket. She selects the sandwiches at random from those on display. The probability that a sandwich is suitable for vegetarians is independently 0.3 for each sandwich.
(a) Find the probability that, of the eight sandwiches, the number suitable for vegetarians is:
(i) 2 or fewer;
(ii) exactly 2;
(iii) more than 3.
Q.7
10 patients are treated with an intervention that is successful 75% of the time.
(a) What is the probability of observing no successes among the ten treatments?
(b) What is the probability of observing 9 or more successes?
Q. 8
Let X~Bern(p). Prove that mean and variance of Bernoulli distribution are p and pq respectively.
Q.9.
Moment generating function, MX(t), is used to find all possible moments (E(Xn)). This function can be used to find mean, variance, and higher order moments. MX(t) can be defined as:
MX(t)=XΣ etx[p(x)] where p(x) is the pmf of any discrete distribution. Use this relation to prove that
(a) For Bernoulli distribution,
MX(t)=pet+q Where q=1-p
(b) For Binomial distribution ,
MX(t)=[pet+q]n.