Binomial mean and variance | CAIE 9709 Probability & Statistics 1 Notes

Chapters

Probability & Statistics 1

Mathematics9709

Mean gives the centre; variance gives the spread

B(20, 0.4): centre and spread

mean = 8 ↓

048121620

mean
np = 8

variance
npq = 4.8

s.d.
√4.8 ≈ 2.19

The mean locates the long-run centre. Variance and standard deviation describe the spread around that centre.

E(X)=np

Symbols

$E(X)$ = long-run mean of X (same as X)
$n$ = number of trials (none)
$p$ = success probability (none)

\operatorname{Var}(X)=npq

Symbols

$\operatorname{Var}(X)$ = variance of X (squared units of X)
$q$ = failure probability, 1-p (none)

Standard deviation returns the spread to the same units as $X$ , so take the positive square root:

\operatorname{SD}(X)=\sqrt{npq}

Worked example

B(80, 0.3)

E(X)=80(0.3)=24

\operatorname{Var}(X)=80(0.3)(0.7)

\operatorname{SD}(X)=\sqrt{16.8}

Answer

E(X)=24

\operatorname{Var}(X)=16.8

\operatorname{SD}(X)=4.10

The ratio variance ÷ mean reveals q

When the mean and variance are given, do not start with two unrelated equations. Dividing cancels $n$ and $p$ :

\frac{\operatorname{Var}(X)}{E(X)}=\frac{npq}{np}=q

Find q. Divide the variance by the mean.
Find p. Use $p=1-q$ .
Find n. Rearrange $np=E(X)$ , then check that $n$ is a whole number.

Worked example

Recover the model, then use it

Suppose $E(X)=6$ and $\operatorname{Var}(X)=4.5$ .

q=\frac{4.5}{6}=0.75

p=0.25,\qquad n=24

Now the recovered model can answer an exact probability.

P(X=5)=p_5

p_5=\binom{24}{5}p^5q^{19}

Answer

P(X=5)=0.176

Common mistake

Standard deviation is

\sqrt{npq}

, not

npq

. Square a given standard deviation before using it as the variance.

Finish with every requested quantity

After recovering $n$ and $p$ , return to the question. It may also ask for a standard deviation or a probability.

Your turnIndependent check [5 marks]

A binomial random variable has mean 8 and variance 4.8. Find $p$ , $n$ and the standard deviation.

Show worked answer

q=\frac{4.8}{8}=0.6

p=1-q=0.4

n=\frac{8}{0.4}=20

\operatorname{SD}(X)=\sqrt{4.8}

Answer

p=0.4

n=20

\operatorname{SD}(X)=2.19

Mean gives the centre; variance gives the spread

B(20, 0.4): centre and spread

mean = 8 ↓

048121620

mean
np = 8

variance
npq = 4.8

s.d.
√4.8 ≈ 2.19

The mean locates the long-run centre. Variance and standard deviation describe the spread around that centre.

E(X)=np

Symbols

$E(X)$ = long-run mean of X (same as X)
$n$ = number of trials (none)
$p$ = success probability (none)

\operatorname{Var}(X)=npq

Symbols

$\operatorname{Var}(X)$ = variance of X (squared units of X)
$q$ = failure probability, 1-p (none)

Standard deviation returns the spread to the same units as $X$ , so take the positive square root:

\operatorname{SD}(X)=\sqrt{npq}

Worked example

B(80, 0.3)

E(X)=80(0.3)=24

\operatorname{Var}(X)=80(0.3)(0.7)

\operatorname{SD}(X)=\sqrt{16.8}

Answer

E(X)=24

\operatorname{Var}(X)=16.8

\operatorname{SD}(X)=4.10

The ratio variance ÷ mean reveals q

When the mean and variance are given, do not start with two unrelated equations. Dividing cancels $n$ and $p$ :

\frac{\operatorname{Var}(X)}{E(X)}=\frac{npq}{np}=q

Find q. Divide the variance by the mean.
Find p. Use $p=1-q$ .
Find n. Rearrange $np=E(X)$ , then check that $n$ is a whole number.

Worked example

Recover the model, then use it

Suppose $E(X)=6$ and $\operatorname{Var}(X)=4.5$ .

q=\frac{4.5}{6}=0.75

p=0.25,\qquad n=24

Now the recovered model can answer an exact probability.

P(X=5)=p_5

p_5=\binom{24}{5}p^5q^{19}

Answer

P(X=5)=0.176

Common mistake

Standard deviation is

\sqrt{npq}

, not

npq

. Square a given standard deviation before using it as the variance.

Finish with every requested quantity

After recovering $n$ and $p$ , return to the question. It may also ask for a standard deviation or a probability.

Your turnIndependent check [5 marks]

A binomial random variable has mean 8 and variance 4.8. Find $p$ , $n$ and the standard deviation.

Show worked answer

q=\frac{4.8}{8}=0.6

p=1-q=0.4

n=\frac{8}{0.4}=20

\operatorname{SD}(X)=\sqrt{4.8}

Answer

p=0.4

n=20

\operatorname{SD}(X)=2.19