Question

Question 1

The probability that a teacher will see 0, 1, 2, 3, or 4 students

(a)        What is the probability that the teacher see 3 students?                                              

(b)       What is the probability that the number of students the teacher will see is between 1 and 3 inclusive?         

(c)        What is the expected number of students that the teacher will see?

(d)       What is the standard deviation?

Question 2

The probability that a house in an urban area will be burglarized is 5%. A sample of 50 houses is randomly selected to determine the number of houses that were burglarized.

(a)        Define the variable of interest, X.                                                      

(b)       What are the possible values of X?                                                    

(c)        What is the expected number of burglarized houses?                                    

(d)       What is the standard deviation of the number of burglarized houses?                      

(e)        What is the probability that none of the houses in the sample was burglarized?

Question 3

A sales firm receives an average of three calls per hour on its toll-free number. Suppose you were asked to find the probability that it will receive at least three calls, in a given hour:

(a) (i)   which distribution does this scenario fit and why?                           

    (ii) define the variable of interest, X.                                                       

    (iii) what are the possible values of X?                                                     

(b)       What is the probability that in a given hour it will receive at least three calls?

Answer 1

Question 1.

The distribution table is missing. But below is a clear explanation on how to solve this.

The probability distribution for x is as shown below.

x	0	1	2	3	4
P(x)	a	b	c	d	e

a)

Probability that the teacher see 3 students P(x=3)=d (In your question there will be specific values for a,b,c,d, and e just replace those values ).

b)

Probability that the number of students the teacher will see is between 1 and 3 inclusive = P(x=1) +P(x=2) +P(x=3) = b+c+d

c)

expected number of students that the teacher will see E[x] = (0*a) +(1*b) +(2* c)+(3*d)+(4*e) =

d)

Question 2.

a) The variable of interest is the number of bulgarized houses in the given sample.

b) The possible values of x are 0,1,2,3,4,5,...,50.

c) Expected number of burglarized houses = np where n=50 and p=0.05 we get E[x] = 50*0.05 = 2.5

d) Standard deviation is sqrt(n*p*(1-p)) = sqrt( 50*0.05*0.95) = 1.5411

e)

It follows a binomial distribution so

Probability that none of the houses in the sample was burglarized

Question 3

a) Poisson distribution is used to find the number of events occur in a particular interval with a fixed mean rate. Here 3 events to be occured at given hour with average rate of 3. Therefore the best fit distribution for this scenario is poisson distribution.

b) The variable of interest is number of phone calls received in a given hour.

c) The probability mass function for the poisson distribution is as shown below.

Probability that in a given hour it will receive at least three calls

answer is 0.5768.

Note:

As you had not uploaded the probability distribution function table for the first question, I assume it as a,b,c,d and e. So please compare with ur question values and replace them to get the answers. and if any query please comment below.