Question

8. Elena just got engaged to be married. She posts a message about the engagement on Facebook. Three of her friends, Alicia, Barbara, and Charlene, will click “like" on her post. Use X, Y, and Z (respectively) to denote the waiting times until Alicia, Barbara, and Charlene click “like" on this post, and assume that these three random variables are independent. Assume each of the random variables is an Exponential random variable that has an average of 2 minutes.

8a. Find P(X<1).

8b. Use your answer to 8a to find the probability that all 3 friends “like" the post within 1 minute.

8c. Use your answer to 8a to find the probability that none of the 3 friends “like" the post within 1 minute.

8d. Use your answer to 8a to find the probability that exactly 1 of the 3 friends “likes" the post within 1 minute.

8e. Use your answer to 8a to find the probability that exactly 2 of the 3 friends “like" the post within 1 minute.

8f. Let V denote the number of friends (among these 3) who “like" the post within 1 minute. Then V is a discrete random variable. What kind of random variable is V? [Hint: In 8b, we have P(V=3); in 8c, we have P(V=0); in 8d, we have P(V=1); in 8e, we have P(V=2). Your answers in 8b, 8c, 8d, 8e should sum to 1.]

a.Bernoulli random variable

b.Binomial random variable

c.Geometric random variable

d.Poisson random variable

Answer 1

Here by the problem,

Elena just got engaged to be married. She posts a message about the engagement on Facebook. Three of her friends, Alicia, Barbara, and Charlene, will click “like" on her post.

Now X, Y, and Z respectively denote the waiting times until Alicia, Barbara, and Charlene click “like" on this post. We further assume that these three random variables are independent and each of the random variables is an Exponential random variable that has an average of 2 minutes.

Then each of X, Y and Z has the pdf,

8a) Then,

.

Let us assume V be the random variable that the total number of friends who hit like within 1 min of post. Here we notice that if we consider each friend hitting like within 1 min as success then each of such cases as Bernoulli trial and as they are independent hence V is the sum of success of 3 Bernoulli trials with probability of success 0.3935. SO V~Binomial(3,0.3935)

8b) Now the probability that all 3 friends “like" the post within 1 minute.

8c. Then the probability that none of the 3 friends “like" the post within 1 minute.

8d) The probability that exactly 1 of the 3 friends “likes" the post within 1 minute.

8e) Lastly the probability that exactly 2 of the 3 friends “like" the post within 1 minute.

8f) Now as earlier discussed V denote the number of friends (among these 3) who “like" the post within 1 minute. Then V is a discrete random variable.

Then by the definition given in the theory V is a random variable following Binomial distribution with parameter (3,0.3935)

Hence option B is the correct one.

Hence the answer..........

Thank you..............