Binomial Distribution Deduction of Ek and Var(k Binomial distribution mean deduction:From the equation k=X1+…+Xm where all XiE {0,1}are Bernoulli distributed random variables with(Xi=1)'s probability equals to P,then:E[Xil=P We get: E[W=E[X+…+X=EX]+…+EX=P+·+卫=nP n times Binomial distribution variance deduction:All Xi are independently Bernoulli distributed random variables.Since Var(Xi)=P(1-P),we get: Var(k=ar(X+…+Xn)=Var(X)+…+Var(Xn) nVar(X1)=nP(1-P) 7/45Binomial Distribution Deduction of E[k] and Var(k) ▶ Binomial distribution mean deduction: From the equation k = X1 + · · · + Xn where all Xi ∈ {0, 1} are Bernoulli distributed random variables with (Xi = 1)’s probability equals to P, then: E[Xi ] = P ▶ We get: E[k] = E[X1 + · · · + Xn] = E[X1] + · · · + E[Xn] = P + · · · + P | {z } n times = nP ▶ Binomial distribution variance deduction: All Xi are independently Bernoulli distributed random variables. Since Var(Xi) = P(1 − P), we get: Var(k) = Var(X1 + · · · + Xn) = Var(X1) + · · · + Var(Xn) = nVar(X1) = nP(1 − P) 7 / 45