Question

In: Statistics and Probability

A store has N clients per day, where the probability that N will be three is...

A store has N clients per day, where the probability that N will be three is 0.1, that N will be two is 0.4, that N will be one is 0.3. The store never gets more than three clients per day.

(a) Is N binomial? Poisson?

(b) Write the cumulative distribution function for N.

(c) What is the average number of clients per day?

(d) You want to study how many bags of milk each client buys. Half of them buy two bags, a quarter buy 1 bag, and the rest buy none. Let X be the number of bags of milk purchased on a given day. Are X and N independent?

(e) What is the probability that 5 bags will be purchased?

(f) What is the probability that there will be 3 clients and that 5 bags will be purchased?

(g) What is the probability the 5 bags will be purchased given that there are three clients?

(h) Find the probability function of X.

Expert Solution

Let N =x be the clients visit each day. Here is the list of probabilities of visiting N=x clients each day.

N=x	P(x)	x* P(x)
3	0.1	0.3
2	0.4	0.8
1	0.3	0.3
0	1-0.1-0.4-0.3=0.2	0

Expected Value of N ,

(a) Is N binomial? Poisson?

Here N is discrete probability distribution which is Poisson distribution, which measures clients count /day.

Poisson distribution counts the number of occurrences in an interval given a certain average occurring rate per interval. While Binomial distribution address problems with a fixed number of events/trials where each event has only two possible events outcome considered as success and failure.

(b) Write the cumulative distribution function for N.

The cumulative distribution function for N , is

N=x	P(x)	CDF : P(N<=x)
0	0.2	0.2
1	0.3	0.3+0.2=0.5
2	0.4	0.4+0.5=0.9
3	0.1	0.9+0.1=1

*CDF = Cumulative distribution function.

(c) What is the average number of clients per day?

Average number of clients / day = Expected value ( Clients visit a day * Probability of visiting that many clients)

From above table , describing various probabilities of clients,

Expected Value of N ,

Hence, average number of clients per day = 1.40.

(d) You want to study how many bags of milk each client buys. Half of them buy two bags, a quarter buy 1 bag, and the rest buy none. Let X be the number of bags of milk purchased on a given day. Are X and N independent?

X: bags of milk purchased , N : clients visited.

Milk bags sold each day is dependent variable, as number of bags of milk purchased (x) is direct depends on the number of clients ( N) visited on a given day.

orchestra answered 1 year ago

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