Independent sums and sample means | Year 12

01 / LEARN

The idea, step by step

Combine independent measurements

Means add when variables are added. For independent variables, variances also add; subtracting an independent variable still adds its variance because the coefficient is squared. Standard deviations do not add.

A linear combination of independent normal variables is normal. Independence is an assumption about how observations are obtained, not a consequence of each marginal distribution being normal. Without it, covariance can change the result.

E (a X + b Y) = a μ_{X} + b μ_{Y}, Var (a X + b Y) = a^{2} σ_{X}^{2} + b^{2} σ_{Y}^{2}

Average independent observations

For n independent observations with the same mean and variance, their sum has mean n mu and variance n sigma squared. Dividing the sum by n gives a sample mean with mean mu and variance sigma squared divided by n.

If the population is normal, the sample mean is exactly normal for every positive sample size. If it is not, a sufficiently large random sample may allow a normal approximation. Multiplying one observation by n is not the same random experiment as adding n independent observations.

\bar{X} \sim N (μ, \frac{σ^{2}}{n}) for independent normal observations

A common trap

Adding standard deviations, or treating nX as n independent observations.

Instead: Add independent variances and identify whether observations are distinct.

02 / WORKED EXAMPLES

See the thinking

Example 1

A normal difference

Independent $X \sim N (10, 4)$ and $Y \sim N (5, 9)$ . Find the distribution of X-Y.

The mean is $10 - 5 = 5$ .
Variance is $4 + 9 = 13$ .
Independence and normality give a normal difference.

X - Y \sim N (5, 13)

Example 2

A small sample from a normal population

Four independent values each follow $N (10, 4)$ . Find the distribution of their mean.

The mean remains 10.
Variance is $4 / 4 = 1$ .
The population is normal, so this is exact even for n=4.

\bar{X} \sim N (10, 1)

Check your understanding

Independent $X \sim N (10, 4)$ and $Y \sim N (5, 9)$ . Find Var(2X-Y).

Show answer

25.

$2^{2} (4) + {(- 1)}^{2} (9) = 25$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Independent $X \sim N (10, 4)$ , $Y \sim N (5, 9)$ . Find the distribution of X+Y.
2 marks
A normal population has mean 80 and SD 12. Find the distribution of a mean from 9 independent observations.
2 marks
For that sample mean, find P(mean>84).
2 marks
If X has variance 4, compare Var(2X) with the variance of X1+X2 for independent copies.
2 marks
Two normally distributed processing times come from the same machine cycle and are strongly related. Can you automatically add their variances?
2 marks
Independent normal X,Y have means 20,12 and variances 9,4. Find P(X-Y<5).
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Independent $X \sim N (10, 4)$ , $Y \sim N (5, 9)$ . Find the distribution of X+Y.
2 marks
Means add and independent variances add.
$N (15, 13)$ .
A normal population has mean 80 and SD 12. Find the distribution of a mean from 9 independent observations.
2 marks
The variance is $144 / 9 = 16$ .
$N (80, 16)$ .
For that sample mean, find P(mean>84).
2 marks
Standard error is 4; $z = (84 - 80) / 4 = 1$ .
0.1587.
If X has variance 4, compare Var(2X) with the variance of X1+X2 for independent copies.
2 marks
$Var (2 X) = 4 (4) = 16$ .
Independent sum variance is $4 + 4 = 8$ .
16 versus 8.
Two normally distributed processing times come from the same machine cycle and are strongly related. Can you automatically add their variances?
2 marks
Dependence may produce nonzero covariance; independence was not established.
No.
Independent normal X,Y have means 20,12 and variances 9,4. Find P(X-Y<5).
3 marks
$X - Y \sim N (8, 13)$ .
Standardise 5 using SD $\sqrt{13}$ .
0.2027.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: continuous random variables, normal distributions and sampling distributions of sample means. Original lessons and practice.

Official curriculum source ↗

Back to Continuous probability and normal distributions