#1160) Calls to the Suicide Hot Line in Gotham City arrive at the rate of 28 cal
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Question
#1160) Calls to the Suicide Hot Line in Gotham City arrive at the rate of 28 calls per day-every day-measured in continuous time. The period from. 8 pm until 4 ann is called the "Lonley Hours." (In this case, we define a "day" to run 24 hours from 8 am until 8 am.) (a) Formulate a probabilistic model for this situation, and assume in all that follows that your model is accurate. (b) Evaluate algebraically the probability of exactly 6 calls between 8 pm and 4 am during tomor- row's Lonely Hours. (c) Let T denote the number of calls that will arrive during a tyical week. Name the distribution of T. Determine the mean and standard deviation (or, arrow length) of the distribution of T. Draw a careful picture of the distribution of T in which you label the exact horizontal coordinates of the 5 important points in the approximate distribution of T as well as the associated areas. (d) Given only that 204 observations occur during the first week, name the distribution (include the numerical values of its parameters) of the subtotal Tz of these 204 that actually occur during the Lonley Hours. Replace the phrase .d uring the first week.. in part (d) by the phrase ..during the first two weeks... and then answer that same question. e that you are now told the one additional fact that 124 of the 204 observations alluded toin part (d) occurred during the Lonely Hours. So, given only that T 204 and TL = 124, name the conditional distribution of the number To that occurred during the "Daytime Hours" from 8 Hint: Revisualize!!! f) Suppos m until 8 pm. (Include the parameter values of this distribution.) (a Name the distribution of the number of days until (for the first time) exactly 6 calls arrive during tExplanation / Answer
Here we use the poisson distribution:
The number of successes in a Poisson experiment is referred to as a Poisson random variable. A Poisson distribution is a probability distribution of a Poisson random variable.For example, suppose we know that a receptionist receives an average of 1 phone call per hour. We might ask: What is the likelihood that she will get 0, 1, 2, 3, or 4 calls next hour. If we treat the number of phone calls as a Poisson random variable, the various probabilities can be calculated.
a)The probability model is:
X=Number of phone calls per hour.X follows Poisson(m),
where m=mean no of phone calls per hour= 28/24 =1.166
b)By Additive property of Poisson Distribution,
If Y=Number of calls between 8 pm and 4 am,then Y follows Poisson(8*m), as the time span is of 8 hours length.
So, Reqd Prob= P(Y=6)=0.08094, by pmf of poisson distribution.
c)By Additive property of Poisson Distribution,
If Z=Number of calls between a full week, Then, Z follows Poisson(24*7*m), as the time span is of 24*7 hours length. So, Z follows Poisson(196) distribution.
Mean=196, SD=14, as P(m) has mean m and variance m.
Picture: Refer to Poisson pmf: https://en.wikipedia.org/wiki/Poisson_distribution.
d) Here the relevant property is :
Suppose that X1 and X2 are independent Poisson distributed random variables with means 1 and 2. Given is that W=X1+X2 is also Poisson distributed with mean 1+2. Then the conditional distribution of X1, given that W=w, is a binomial distribution with n=w and p=1/(1+2) .
Here 1/(1+2)= Proportion of Lonely Hour in a day =8hr/24hr= 1/3
So, HereTL follows Binomial distribution with n=204 and p=1/3.
(First four parts answered)
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