Answer to Question #234779 in Statistics and Probability for Booblex

Question #234779

b) Phone calls enter the ”support desk” of an electricity supplying company on the average two every 3 minutes. If one assumes an approximate Poisson process:

(i) What is the probability of no calls in 3 minutes?

(ii) What is the probability of utmost 6 calls in a 9 minute period?

[2 marks] [2 marks]

c) Suppose the earnings of a laborer, denoted by X, are given by the following probability distribution.

 P(xi) 0 8/27

xi

 1 12/27

2 6/27 3 1/27

Find the laborer’s expected earnings and the variance of his earnings.


1
Expert's answer
2021-09-09T00:47:15-0400

Let "X=" the number of calls: "X\\sim Po(\\lambda t)."

(i)


"\\lambda t=2"

"P(X=0)=\\dfrac{e^{-\\lambda t}(\\lambda t)^0}{0!}=e^{-2}\\approx0.135335"

(ii)


"\\lambda t=6"

"P(X\\leq6)=P(X=0)+P(X=1)+P(X=2)"

"+P(X=3)+P(X=4)+P(X=5)+P(X=6)"

"=\\dfrac{e^{-\\lambda t}(\\lambda t)^0}{0!}+\\dfrac{e^{-\\lambda t}(\\lambda t)^1}{1!}+\\dfrac{e^{-\\lambda t}(\\lambda t)^2}{2!}+\\dfrac{e^{-\\lambda t}(\\lambda t)^3}{3!}"

"+\\dfrac{e^{-\\lambda t}(\\lambda t)^4}{4!}+\\dfrac{e^{-\\lambda t}(\\lambda t)^5}{5!}+\\dfrac{e^{-\\lambda t}(\\lambda t)^6}{6!}"

"=e^{-6}(1+6+18+36+54+64.8+64.8)"

"=224.6e^{-6}\\approx0.606303"

c)


"\\begin{matrix}\n x_i & 0 & 1 & 2 & 3\\\\\n p(x_i) & 8\/27 & 12\/27 & 6\/27 & 1\/27\n\\end{matrix}"

Check


"\\dfrac{8}{27}+\\dfrac{12}{27}+\\dfrac{6}{27}+\\dfrac{1}{27}=1"

"E[X]=\\dfrac{8}{27}(0)+\\dfrac{12}{27}(1)+\\dfrac{6}{27}(2)+\\dfrac{1}{27}(3)=1"

"E[X^2]=\\dfrac{8}{27}(0)^2+\\dfrac{12}{27}(1)^2+\\dfrac{6}{27}(2)^2+\\dfrac{1}{27}(3)^2=\\dfrac{5}{3}"

"Var(X)=E[X^2]-(E[X])^2=\\dfrac{5}{3}-(1)^2=\\dfrac{2}{3}"


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