Question

In: Statistics and Probability

Let X denote the number of Canon SLR cameras sold during a particular week by a...

Let X denote the number of Canon SLR cameras sold during a particular week by a certain store. The pmf of X is

x 0 1 2 3 4

pX(x) 0.1 0.2 0.3 0.25 0.15

Seventy percent of all customers who purchase these cameras also buy an extended warranty. Let Y denote the number of purchasers during this week who buy an extended warranty.

(a) What is P(X = 4, Y = 2)? [Hint: This probability equals P(Y = 2|X = 4) · P(X = 4); now think of the four purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended warranty.] (Round your answer to four decimal places.)

P(X = 4, Y = 2) =

(b) Calculate P(X = Y). (Round your answer to four decimal places.)

P(X = Y) =

(c) Determine the joint pmf of X and Y. y x (0.7)y(0.7)x−y · pX(x) x y (0.7)y(0.3)x−y · pX(x) x y (0.7)x(0.3)x−y · pX(x) y x (0.7)x(0.7)x−y · pX(x)

Determine the marginal pmf of Y. (Round your answers to four decimal places.)

y 0 1 2 3 4

pY(y)

Please show work.

Expert Solution

Answer A:

P(X =4,Y =2) = P(Y =2|X=4)/P(X=4)

P(X= 4) = 0.15

Now P(Y=2|X= 4) is the binomial distribuiton wiht success is 0.7 and failure is 0.3

Asper binomial distribuiton P(2,0.7) = ⁴C₂ * 0.7^2 *0.3^2

=6*0.49*0.09

=0.2646

SO

P(X =4,Y =2) = 0.15 *0.2646

= 0.03969

So out of 4 camera sold, 0.03969 probability that they will find the extended warenty for 2 cameras

Answer B:

P(X=Y) indicates that probability of all the camera should have the extended warenty

P(X =Y) = P(X=0,Y=0) + P(X=2,Y=2) + P(X=3,Y=3) + P(X=4,Y=4) + P(X=1,Y=1)

Now

P(Y = 0|X =0)*P(X=0) = 1*0.1 = 0.1 (if person don't buy camera they will definately not buy the warrenty

P(Y = 1|X =1)*P(X=1) = 0.7*0.2 = 0.14 (if person buy 1 camera than probability that he will buy warenty is 0.7)

P(Y = 2|X =2)*P(X=2) = binom(2,2,0.7)* 0.3 = 0.7^2*0.3 =0.147

P(Y = 2|X =3)*P(X=3) = binom(3,3,0.7)* 0.25 = 0.7^3*0.25 =0.08575

P(Y = 2|X =4)*P(X=4) = binom(4,4,0.7)* 0.15 = 0.7^4*0.15 =0.036015

So P(X=Y) = 0.1 + 0.14 + 0.147 + 0.08575 +0.036015 = 0.508765

So probability that all the purchased camera having extended warrenty is 0.508765

orchestra answered 1 year ago

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