ASSIGNMENT OF PROBABILITY AND STATISTICS

 ASSIGNMENT OF PROBABILITY AND STATISTICS

Question No. 1 Three machines A, B and C produce 60%, 30% and 10% respectively of the total number of items of a factory. The percentage of defective output of these machines are respectively 2%, 3% and 4%. An item is selected at random and is found to be defective. Find the probability that the item was produced by machine C.

Solution: Let P(A), P(B) and P(C) denote the probabilities of choosing an item produced by A, B, C respectively. Also, let P(X/A), P(X/B), P(X/C) denote the probabilities of choosing a defective item from the outputs of A, B, C respectively. Then, from what is given, we have

        

                P(A)= 60/100 = 0.60  P(B) = 30/100 =0.30 P(C) = 10/100=0.10

      

            P(X/A) =2/100= 0.02    P(X/B) =3/100= 0.03 P(X/C) =4/100 =0.04

We are required to find P(C/X), the probability that the item selected is produced by machine C given that the item is defective. By Baye’s theorem, we find that

       

            P(C/X)= P(X/C) P(C)/ P(A) P(X/A) +P(B) P(X/B) +P(C) P(X/C)

                        = (0.04) (0.10)/ (0.60) (0.02) +(0.30) (0.03) + (0.10) (0.04)

                       = 0.16

Question No.2  Find the constant k so that Is a joint probability density function. Are X and Y independent?

Solution: We observe that f(x, y)  0 for all x, y; if k  0.

Further,

            

Accordingly, f(x, y) is a joint probability density function if k = 2/3 with k = 2/3, we find that the marginal density functions are

             

               We observe that g(x) h(y) = f(x, y)  X and Y are independent random variables.

Question No.3

Question No 4

Graphical representation of trend line relating to the production of wheat by the method of free hand curve is shown in figure

Question No. 5 Let X be a random variable and its probability mass function is Find the m.g.f. of X and hence it’s mean and variance.

Question No.6 The diastolic blood pressures of men are distributed as shown in table. Find the standard deviation and variance.

Standard deviation =  = 2.336 (mm)

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