Let X B(n,p). **Metric** | **Value** | **Meaning** Expectation | mu=E(X)=np | Central value of the distribution. Variance | sigma2=Var(X)=np(1-p) | Squared measure of spread. Standard deviation | sigma=SD(X)=sqrt(np(1-p)) | Measure of spread in the units of (X). Mode | The x for which P(X=x) is greatest. | Most probable value of (X). Source strategy for finding the mode - Find the expectation mu=E(X)=np. - If mu is an integer, compare the probabilities at mu-1,mu,mu+1. - If mu is not an integer, compare the probabilities at the two nearest integers and. - The candidate with the greatest probability is the mode. Illustration For X B(6,0.6), mu=6(0.6)=3.6. Compare P(X=3) and P(X=4). P(X=3)=0.27648, P(X=4)=0.31104 Therefore, the mode is 4. The worked examples in this unit show an alternative method for finding the mode of a binomial distribution.

If \(\mu\) is an integer, compare the probabilities at \(\mu-1,\mu,\mu+1\).

If \(\mu\) is not an integer, compare the probabilities at the two nearest integers \(\lfloor\mu\rfloor\) and \(\lceil\mu\rceil\).

The candidate with the greatest probability is the mode.

For \(X\sim B(6,0.6)\), \(\mu=6(0.6)=3.6\). Compare \(P(X=3)\) and \(P(X=4)\).

Calculator comparison

The worked examples in this unit show an alternative method for finding the mode of a binomial distribution.

Binomial Metrics and Problem-Solving Strategies

Binomial Metrics and Problem-Solving Strategies

Metric	Value	Meaning
Expectation	\(\mu=E(X)=np\)	Central value of the distribution.
Variance	\(\sigma^2=\operatorname{Var}(X)=np(1-p)\)	Squared measure of spread.
Standard deviation	\(\sigma=\operatorname{SD}(X)=\sqrt{np(1-p)}\)	Measure of spread in the units of (X).
Mode	The \(x\) for which \(P(X=x)\) is greatest.	Most probable value of (X).