Bing has a beautiful baby named Nora. She is very attentive to her baby\'s needs
ID: 2934764 • Letter: B
Question
Bing has a beautiful baby named Nora. She is very attentive to her baby's needs and has purchased a speciali monitor so she can always hear and see her daughter. When the baby cries, Bing instantly responds. Bing realizes that her baby cries on average 3 times in 4 hours. Assume each cry is independent. a) Let C be the number of cries between 8:00-10:00 AM. What are the distribution, parameter(s), and support b) c) d) e) of C? What is the probability there are 2 cries between 8:00 AM-10:00 AM and 4 cries between 1:00 PM-4:00 PM? Bing responded to Nora's cry at 3:45 PM. What is the probability that the next cry is after 5:45 PM? What distribution and parameter(s) are you using? Bing has been timing the cries. Given that it has been over 30 minutes since Nora last cried, what is the probability that the next cry will come in less than 45 minutes from Nora last cried? There were 15 cries in the past 24 hours. What is the probability that two of these cries were between 3:00 AM and 4:30 AM?Explanation / Answer
Given 3 cries in 4 hours => average number of cries per hour = ¾ = 0.75.
Part (a)
Given C = number of cries between 8:00 -10:00 AM,
C is distributed as Poisson ANSWER 1
The parameter
=
= average number of cries in 2hours (8:00 -10:00 AM)
= 2 x 0.75
= 1.5 ANSWER 2 [average number of cries per hour = 0.75 => average number of cries in 2 hours = 2 x 0.75, by property of Poisson Distribution.]
Part (b)
Probability of 2 cries between 8:00 -10:00 AM
Let X = number of cries between 8:00 -10:00 AM. Then, X ~ Poisson (1.5). So,
Probability of 2 cries between 8:00 -10:00 AM
= P(X = 2/ = 1.5) = 0.2510 ANSWER 2
Let Y = number of cries between 1:00 PM - 4:00 PM (i.e., 3 hours). Then, Y ~ Poisson (2.25). [2.25 = 3 x 0.75] So,
Probability of 4 cries between 1:00 PM - 4:00 PM
= P(X = 4/ = 2.25) = 0.1125 ANSWER 3
Part (c)
Given last cry was at 3:45 pm, probability that the next cry is after 5:45 pm is equal to probability that there is no cry between 3:45 PM - 5:45 PM i.e., 2 hours
= P(X = 0/ = 1.5) = 0.2231 ANSWER 4
Distribution used here is Poisson with parameter 1.5 ANSWER 5
Part (d)
Given no cry for 30 minutes, probability that the next cry is within next 45 minutes from last cry is equal to probability that there is a cry in 15 minutes.
= P(Z = 1/ = 0.1875) = 0.1554 ANSWER 4 [Z = number of cries during 15 minutes and hence Z ~ Poisson (0.75/4)]
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