Question 4 The Binomial Probability Distribution is given by:\[ f_k = \binom{n}{k} p^k (1-p)^{n-k} \]where \( n \) is an integer with \( n \geq 0 \), \( k \) is an integer with \( 0 \leq k \leq n \), \( p \) is a real number with \( 0 < p < 1 \), and\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]The quantity \( f_k \) 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\[ 1 = \sum_{k=0}^{n} f_k \quad \text{(normalization)} \]\[ np = \sum_{k=0}^{n} k f_k \quad \text{(mean)} \]\[ np(1-p) = \sum_{k=0}^{n} (k-np)^2 f_k \quad \text{(variance)} \]When \( n \gg 1 \), we can use Stirling's Formula to approximate the factorials provided that we avoid the extremes of the distribution, i.e., as long as \( k \gg 1 \) and \( k \ll n \) as well. Show that this approximation leads (not unexpectedly) to the Gaussian Probability Distribution:\[ f_k \rightarrow \frac{1}{\sqrt{2\pi np(1-p)}} \exp \left[ -\frac{(k-np)^2}{2np(1-p)} \right] \]

Basics of Binomial Distribution 1. It's a discrete distribution2. It only counts success or…

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