More on Poisson

x = # ATM uses/hr

λ = 4 uses per hr on average

x~poisson(4)

1. p(x=5) = “probability of 5 uses in next hr”

   recall: P(X = x) = (λ^x * e^(−λ))/x! P(X = 5)=(4⁵ * e^−4)/5!=0.156

2. P(x <= 5)= .785

   just take the summation of all from 5

3. P(X >= 6) = 1 - P(X<=5) -> when goes to infinity take 1-x

   P(X ≥ 6)=1-P(X<5)

   =1-0.785

   = 0.215

Suppose that a person taking Vitamin C supplements contracts an average of three colds per year and that this average increases to five colds per year for persons not taking Vitamin C supplements. Suppose further that the number of colds a person contracts in a year is a Poisson random variable.

X1 = # of colds in a given year for a person taking Vitamin C

X2 = # of colds in a given year for a person NOT taking Vitamin C

X1 ~ Poisson(3), X2 ~Poisson(5)

1. Find the probability of no more than two colds for a person taking Vitamin C supplements and a person not taking Vitamin C supplements in a given year.
   • (X1 <= 2) = P(x1 = 2) + P(x1 = 1) + P(x1 = 0) -> change lambda to 3
   • (X2 <= 2) = P(x2 = 2) + P(x2 = 2) + P(x2 = 0) -> change lambda to 5

2. Suppose 70% of the population takes Vitamin C supplements. Find the probability that a randomly selected person will have no more than two colds in a given year.

   P(x<=2) = P(x<=2 and C) + P(x<=2 and ~C)

   = P(C) * P(X<= 2 | C) + P(~C)*P(X<=2|~C)

   = (0.7)*(0.423) + (0.3)*(0.125)= 0.3336

A = P(coming no later than six p.m) = .8

B = P(Coming no later than six p.m) = .7

P(Choosing A) = .4 P(Choosing B) = .6

1. What proportion of times is amy home no later than 6 pm

   Amy home no later than 6 pm = (A n P(choosing A)) + (B n P(choosing B))

   80/100 * 40/100 + 70/100 * 60/100 = 74/100

   (choosing A) * (A | P(choosing A)) + P

2. if amy is home after 6 pm today what is the probability that she took route B

   P(home after 6 pm) = P(.4 n .2) + P(.6 n .3) = .26

   P(take root B) = .6

   P(home after 6pm | take root B) =

3 copies of boojs bought for = $6

sells each book for 12

unsold copies are returned for 2

let x = {number of copies sold}

let y = {net revenue}

x= 0 y = -18, x= 3 y=18, x=1 y=-8, x=2 y = 24-18 + 2,

0 1 2 3