Bernoulli process. You get text messages at an average rate of 10 texts per minu

Question

Bernoulli process. You get text messages at an average rate of 10 texts per minute. You model this by a Bernoulli process, with discrete time, measured in seconds. Equivalently (in an ideal model), this is 1/6 text per second.

(a) Find the probability that you get exactly 10 texts in 1 minute (simplify your answer as a decimal).Declare your variables. Your solution must contain: “Let X denote the number of texts you get in 1 minute. Then X ????...”.

(b) Find the probability that you get at least 10 texts in 1 minute.

(c) Find the probability that you get the 10th text at exactly the 60th second.

(d) Find the average waiting time in seconds until you get 10 texts. .Declare your variables. Your solution must contain: “Let W denote the number seconds until you get the 10th text. Then W ????...”.

(e) Let t be the answer to (d). Find the probability that you get exactly 10 texts in t seconds.

Explanation / Answer

p(text) = 1/6

p (no text) =5/6

a)

10 text in 1 min

60c10 (1/6)^10 (5/6)^50 = 0.1370131

b)

p(x>10)

= 1-p(x<10) =

1- sum(60ci (1/6)^i (5/6)^(60-i)) = 1-0.5833866= 0.4166134

c)

10th text at the 60th second

p(x=9)= 59C9 (1/6)^9 (5/6)^50

p(x=9)p(x=1) =  59C9 (1/6)^9 (5/6)^50 *(1/6) = 0.02283552

d)

let w be the wait time untill 10 texts

p(x=10 in w sec) =

wc10 (1/6)^10 * (5/6)^(w-10)

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