Question 1207035: The number of births at the local hospital has a Poisson distribution with an average of 48 per day.
(a)
What is the probability distribution for the daily number of births at this hospital?
p(x) =
(b)
What is the probability distribution for the number of hourly births?
p(x) =
(c)
What is the probability that there are fewer than 3 births in a given hour? (Round your answer to three decimal places.)
(d)
Within what interval does Tchebysheff's Theorem suggest you would expect to find the number of hourly births at least 89% of the time? (Round your answer up to the nearest whole number.)
interval (a,b) =
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you can use the poisson distribution calculator at to solve parts (a) through (c).
an alternative is to use the poisson distribution table.
here's a reference on the poisson distribution itself.
https://www.youtube.com/watch?v=zA7fp2s7FlM
here's a link to the calculator.
https://stattrek.com/online-calculator/poisson#google_vignette
here are the results from using the calculator.
you have to set the time interval to be consistent.
you were given 48 births in a 24 hour period.
since you want number per hour, you have to set the average to per hoursly.
48 per 24 hours is equivalent to 2 per hour because the hospital is open 24 hours a day.
so the average is 2 per hour and the variable you are looking for is 3.
you enter 2 for the average and 3 for the number you are looking for and the calculator gives you a bunch of statistics.
since you are looking for x < 3, you choose that.
you get p(x<3) = 0.67668.
part (a) answer is 48.
part (b) answer is 2.
part (c) answer is .67668.
you can also use the poisson dictribution table for cumulative results.
one such table can be found at https://ghaidab.weebly.com/uploads/1/1/2/9/11294132/poisson_cdf_table.pdf
lambda is the mean of the distribution, so for this problem, you would look for a lambda of 2 and a value of x as 2.
you are looking for p(x) < 3 which is the same as p(x) <= 2.
p(x) <= is what the table is set up to look for.
you should find that p(x <= 2) = .6767 from the table.
this agrees with what the calculator provided as p(x) < 3.
if you wanted to find p(x) > 3, you would look for 1 - p(x) <= 3.
that would be equal to 1 minus p(x <= 3) = 1 - .0.85712 = .14288.
if you're using the calculator, it tells you othat straight away.
if you're using the table, you look for p(x) <= 3 and take 1 minus that.
the calculator goes to 5 decimal places.
the table goes to 4.
for part (d), you need to use chebyshev's theorem.
that's the way the internet spells it, i believe.
that theorem states that the smallest probability of any type of distribution is given by the equation of p(|x-m|>=k*s) <= 1 - 1/k^2.
this says that the probability that the variable value minus the mean is greater than or equal to k * the standard deviation is smaller than or equal to 1 - 1/k^2.
k is the number of standard deviations above the mean.
the mean is assumed to be in the middle of the range, so the standard deviation is the same whether above or below the mean.
the absolute value symbol takes care of that by making (x-s) always positive.
in your problem, mean is 2 and variance is also 2, since this is a property of the poisson distribution.
standard deviation is therefore sqrt(2).
at least 89% of the time means the probability that it is greater than or equal to 89% = .89
.89 is the minimum proportion.
the formula is p(|x-m| >= k*s) >= 1 - 1/k^2 = .89
add .89 to both sides of the equation abd add 1/k^2 to both sides of the equation to get 1.89 = 1/k^2.
solve for k^2 to get k^2 = 1/1.89.
solve for k to get k = sqrt(1/1.89) = .72739
the interpretatio9n of that would be that:
the minimum percentage of values that are .72739 * the standard deviation about the mean is equal to .89.
here's a reference on chebyshev's formula.
https://statisticsbyjim.com/basics/chebyshevs-theorem-in-statistics/
the reference states that the maximum proportion = 1/k^2 and the minimum proportion is 1 - 1/k^2.
we used the minimum proportion to solve your problem.
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