Question

3. A new artist stands in the middle of the Hollywood Walk of Fame and randomly asks tourists to buy his/her new CD. The probability of successfully selling a CD is 0.10. Let X be the number of tourists asked to purchase a CD when the first CD is sold. (a) What distribution does X follow? State the name and parameters. (c) What is the variance and standard deviation of the number of attempts needed before selling a CD? (d) What is the probability that the 5th tourist asked is the first to buy the artist's CD? (e) What is the probability that it will take less than 3 tourists before selling another CD? Compute this value using R and state what function you used. Also, compute the value by hand. (f) If the artist only has time to ask 60 tourists, what is the probability that he/she will fail to sell a CD?

Answer 1

Solution Given The probability ofselling a CD is,p 0.10 (a) Let X number of tourists asked to buy the CD before the 1st CD is sold The random variable X follows a geometric distribution with parameter p the probability of success X~Geo (p 0.10) The pdfof geometric distribution is: Р(х -к) - (1- р)* p:k - 0,1,2... Therefore, the distribution ofX is |Geometric(p = 0.10) (b) Compute the expected value ofX- E(x)=1-p 1-0.10 0.10 9 Therefore, the expected number of attempts required before selling the 1st CD is 9

Compute the variance of X- 1-p - (х)л 1-0.10 (oro) = 90 Compute the standard deviation ofX- SD(x)=1-p p 2 /1-0.10 (0.10) V90 =9.486833 9.487 Therefore, the variance and standard deviation of the number of attempts required before selling the 1st CD are 90 and |9.487|respectively. (d) Compute the probability that the first CD is sold to the 5th tourist- Р(х - 5) Use the Excel function "-NEGBINOMDIST(4,1,0.1)" to compute the probability foNEGBINOMDIST(4,1,0.1) D E РX-5) - 0.0656

**A negative binomial distribution with 1 success after k trials is a geometric distribution. Therefore, the probability that the first CD is sold to the 5th tourist is |0.0656 Compute the probability that another CD is sold after failing to sell less than 3 tourists- P(х <3) - Р(х-о) + P(х -1) + P(х -2) _ The probability ofsuccessfully selling the CD is same for each trial The R function to compute the probability is: pgeom (q, prob, lower.tail = TRUE) Here the value of'q' is 3 and the value of 'prob' is 0.1 Compute the value ofP (X<3) by hand- Р(х <3) - Р(х —0) + P(х -1)+ P(х - 2) -(1-0.1)'x0.1)-(1-0.1)x0.1)-(1-0.1)x0.1) _ 0.10.09 0.081 0.2710 Therefore, the probability that another CD is sold after failing to sell less than 3 tourists is 0.2710

(f) Compute the probability that the artist could not sell any CD to the 60 tourist he/she met- P(X60)(1-0.1) (0.1) 0.001797x 0.1 0.0001797 0.00018 Therefore, the probability that the artist could not sell any CD to the 60 tourist he/she met is 0.00018