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.

(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.

Solutions

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.

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.

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 11 months ago

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