Question

Suppose that a company is testing a new product and surveys potential consumers. The people who are surveyed simply choose if they like or dislike the product. Based on the survey results, 70% of the people like the product, while the other 30% does not like the product. Suppose that 4 people are randomly selected from the group of people who are surveyed, and let X denote the number of the people who like the product (a) Write out the probability mass function of X (b) Write out the cumulative distribution of X. (You can just use a) to represent the probability that X-a.) (c) Write out the mean and variance of X (No need to calculate out the final number, just leave the formula there with corresponding numbers plugged in.) (d) Let Y be another random variable such that Y - VX Write out the mean of Y (again, no need to calculate out the final number, just leave the formula there with

Answer 1

Concept:

• The important part here is calculating the probability of x people from 4 selected liking the product, P(X=x)
• For this we need to first choose x people from 4, multiply with probability of x being liking the product and (4-x) not liking the product

$\small P(X=x) = \binom{4}{x}* 0.7^{x} *0.3^{4-x}$

Please find the solution below

X = Number of people who Product über of people who like the X can take values from 0 to L P(x-u) - Ac * (0-7)" * (0-3) So, the PMF would be In to 12 13 14 P(x-wo ooº: • 015 - 2016 -4 11 4 2 1 | Cumulative distribution of x . F(x)=P(x (x) = { Plui) X.in F(0) = P(x(0) - P(x=0) - flo) F(1) = P(XC 1) = flotf (1) FL2) = HO) +f11) +$12) FL3) = flo) + f (1) +f12) + P (3) Flu) = f (0) +$11) +f12) + f23) +f14) F15) at

cdf o f (o) for for ulo oculi F(x) = fol+f(1) for 1 cuc2 f(0) + f(1) + f(2) for2.cn <3 f(0) + f(1) + f (2) + f(3) for 3 cucu for us 4 Mean , E(X) 니 Elu) = {ui plui) = 1 X0:0756 + 2x.2646 +3 x.4116) + 4 *.2401 - varianu & Elu² Plus - Enop (m)] = {1x0.0756) +(4*.2646 +19x14116) +(16*12401) (28) = 0.84

(d) y = 5x g(x) = a E [2C)] = { 2017 P414) u = Eru Pu (n) = 2x0.0756 £ 2 *.2646 + 3 *.4116 +2*.2401 16 yi