Question

Part B: You can also find the standard deviation for the binomial probability distribution of a specific outcome in a binomial experiment. Use the formula to find the standard deviation. You've already identified n and p in Part A. Show your work, and round your answer to two decimal places.

Answer 1

Mean = np + n(n-1)p²

Standard Deviation= √ σ² = √npq

Explanation:

The m.g.f of the binomial probability distribution b(x;n,p) is derived as below.

M₀t = E (e)^tx

= ∑(e)^tx (nCx) (p)^x q^(n-x) { x varies from 0 to n}

= ∑(e)^tx (nCx) (pe^t)^x q^(n-x)

= (q +pe^t)^n

The expansion of this binomial is purely algebraic and need not to be interpreted in terms of probabilities.

WE get the moments by differentiating M₀(t) once, twice, etc. with respect to t and putting t=0

Thus

μ₁` = E(X) = [ d/dt (q +pe^t)^n] t=0

= [ npe^t (q +pe^t)^n-1] t=0

= np

And

μ₂` = E(X²) = [ d²/dt² (q +pe^t)^n] t=0

= [ npe^t (q +pe^t)^n-1] + [ n(n-1)p²e^²t (q +pe^t)^n-2] t=0

= np + n(n-1)p²

Variance= μ₂= μ₂` - μ₁`²

σ² =E(X²)- E(X)²

σ² =np + n(n-1)p²- (np )²

σ² =npq

Standard Deviation= √ σ² = √npq