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4. The Binomial Probability Distribution is given by ()^(---. where n is an integer with n20, k is an integer with 0≤ k ≤n, p is a real number with 0 < p < 1, and n! k!(n-k)! The quantity f gives the probability of obtaining exactly k successes out of n trials, if the probability of success for each trial is p. One can show that 1= Σfk k=0 np = Σkfk k=0 fk → np(1-p) = (k-np)² fx k=0 (normalization) (mean) 1 √2rnp(1-P) When n» 1, we can use Stirling's Formula to approximate the factorials provided that we avoid the extremes of the distribution, i.e., as long as k>1 and kn as well. Show that this approximation leads (not unexpectedly) to the Gassian Probability Distribution: (k - np)² (variance). exp

4. The Binomial Probability Distribution is given by ()^(---. where n is an integer with n20, k is an integer with 0≤ k ≤n, p is a real number with 0 < p < 1, and n! k!(n-k)! The quantity f gives the probability of obtaining exactly k successes out of n trials, if the probability of success for each trial is p. One can show that 1= Σfk k=0 np = Σkfk k=0 fk → np(1-p) = (k-np)² fx k=0 (normalization) (mean) 1 √2rnp(1-P) When n» 1, we can use Stirling's Formula to approximate the factorials provided that we avoid the extremes of the distribution, i.e., as long as k>1 and kn as well. Show that this approximation leads (not unexpectedly) to the Gassian Probability Distribution: (k - np)² (variance). exp

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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4. The Binomial Probability Distribution is given by (2)^(-)--*. where n is an integer with n≥ 0, k is an integer with 0≤ k <n, p is a real number with 0 < p < 1, and n! k!(n-k)! The quantity f gives the probability of obtaining exactly k successes out of n trials, if the probability of success for each trial is p. One can show that 1= Σfk k=0 np = Σkfk k=0 fk → np(1-p) = (k-np)² fx k=0 n-k (normalization) 1 √2rnp(1-P) (mean) When n» 1, we can use Stirling's Formula to approximate the factorials provided that we avoid the extremes of the distribution, i.e., as long as k>1 and kn as well. Show that this approximation leads (not unexpectedly) to the Gassian Probability Distribution: (k - np)² exp (variance).
Transcribed Image Text:4. The Binomial Probability Distribution is given by (2)^(-)--*. where n is an integer with n≥ 0, k is an integer with 0≤ k <n, p is a real number with 0 < p < 1, and n! k!(n-k)! The quantity f gives the probability of obtaining exactly k successes out of n trials, if the probability of success for each trial is p. One can show that 1= Σfk k=0 np = Σkfk k=0 fk → np(1-p) = (k-np)² fx k=0 n-k (normalization) 1 √2rnp(1-P) (mean) When n» 1, we can use Stirling's Formula to approximate the factorials provided that we avoid the extremes of the distribution, i.e., as long as k>1 and kn as well. Show that this approximation leads (not unexpectedly) to the Gassian Probability Distribution: (k - np)² exp (variance).
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