Bing has a beautiful baby named Nora. She is very attentive to her baby’s needs

Question

Bing has a beautiful baby named Nora. She is very attentive to her baby’s needs and has purchased a special 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 of C?

b) 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?

c) 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?

d) 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?

e) 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

Ans:

a)Poisson distribution with mean=3 per 4 hrs=0.75 per hr

b)probability there are 2 cries between 8:00 AM-10:00 AM and 4 cries between 1:00 PM-4:00 PM

=e-1.5*(1.52/2!)*e-2.25*(2.254/4!)

=0.0283

c)Exponential distribution with rate=0.75 per hr

P(x>2 hr)=e-0.75*2=e-1.5=0.2231

d)As,exponential distribution is memoryless,it will be equivalent to

P(x<15 min)=P(x<0.25 hr)=1-e-0.75*0.25=1-e-0.1875=1-0.8290=0.171

e)Memoryless property

probability that two of these cries were between 3:00 AM and 4:30 AM

=e-0.75*1.5*((0.75*1.5)2/2!)=e-1.125*((1.125)2/2!)=0.2054

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