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NSW Y12 Maths - Extension 1 Binomial Distribution Mean and Variance of a Binomial Distribution

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Mean and Variance of a Binomial Distribution Theory

Given that \(X \sim B(n, p)\) then the mean \(\mu=E(x)=n p\) and the variance \(\sigma^2=\operatorname{var}(x)=n p(1-p)\).\\ The standard deviation \(\sigma=\sqrt{n p(1-p)}\)\\ \textbf{Example 1}\\ For \(X \sim B\left(6, \dfrac{1}{4}\right)\) find\\ (i) \(E(x)\)\\ (ii) \(\operatorname{Var}(x)\)\\ (iii) \(\sigma\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \text { (i) } \quad E(x)=n p & =6 \times \frac{1}{4} \\ & =1.5 \end{aligned}$\\ $\begin{aligned} \text { (ii) } \quad \operatorname{Var}(x) & =np(1-p) \\ & =6 \times \frac{1}{4} \times \frac{3}{4} \\ & =1.125 \end{aligned}$\\ $\begin{aligned} \text { (ii) } \quad \sigma & =\sqrt{1.125} \\ & =1.06 \end{aligned}$\\

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  • Mean and Variance of a Binomial Distribution - Video - Binomial distribution expected value variance and standard deviation

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