STAT 400 Chap.7 Sums of Random Variables and the Central Limit Theorem
Sums of Random Variables and the Central Limit Theorem
What always adds and what only sometimes survives
For any random variables, the mean of a sum is the sum of the means. For independent variables, the variance of a sum is the sum of the variances. Those two facts hold regardless of the distributions involved.
The shape of a sum is harder, but the normal family is special: any linear combination of independent normal variables is normal, with the mean and variance given by the usual rules.
The mgf explains why, since the mgf of an independent sum is the product of the individual mgfs and products of normal mgfs are normal mgfs.
The sample mean
Applying those rules to a random sample of size n gives the two most used results of the course: the sample mean has expected value μ and variance σ²/n. Its standard deviation, σ/√n, is called the standard error.
For a normal population the sample mean is exactly normal. Because precision grows only with the square root of n, quadrupling a sample halves the standard error.
The central limit theorem
The central limit theorem says that for a random sample from any population with finite variance, the standardised sample mean approaches a standard normal distribution as n grows. The same holds for the sum.
This is why normal calculations appear in problems about exponential service times, dice, or proportions, none of which are normal themselves.
The speed of convergence depends on the population: symmetric populations converge quickly, while strongly skewed ones need larger samples.
Approximating discrete counts
Since a binomial count is a sum of independent Bernoulli trials, it is approximately normal with mean np and variance np(1 − p) when n is large, commonly checked by np ≥ 5 and n(1 − p) ≥ 5. Because the count lives on integers, each value is treated as a bar of width one, and the event is rewritten with half-unit boundaries before standardising.
This continuity correction often moves the answer in the second decimal place, and the course's objectives ask for it explicitly.
Where this leads
This chapter closes the second content block. Its standard error σ/√n is the building block of every confidence interval and test statistic in the final third of the course.
What this chapter covers
- 01
Means and variances of sums
- 02
Linear combinations of normal variables
- 03
Distribution of the sample mean
- 04
Central limit theorem
- 05
Normal approximation to the binomial
- 06
Continuity correction
Worked example · free
Approximate a count with continuity correction
- 1Check conditions: np = 50 and n(1 − p) = 150, both at least 5.
- 2Mean 50, variance 200(0.25)(0.75) = 37.5, standard deviation 6.124.
- 2Correct the boundary: P(X ≥ 60) becomes P(Y ≥ 59.5), so z = 9.5/6.124 = 1.55.
- 1From the table, 1 − Φ(1.55) = 1 − 0.9394 = 0.0606.
Key terms
- Standard Error
- The standard deviation of a statistic; for the sample mean it equals σ/√n.
- Central Limit Theorem
- The result that the standardised mean of a large random sample from any finite-variance population is approximately standard normal.
- Continuity Correction
- The adjustment of a discrete boundary by one half before using a continuous normal approximation.
- Sampling Distribution
- The probability distribution of a statistic, such as the sample mean, over repeated samples of the same size.
Sums of Random Variables and the Central Limit Theorem FAQ
How large does n need to be for the central limit theorem?
It depends on the population. A symmetric population can look normal at n = 5 or so, while a skewed one needs more; many courses use n of about 30 as a working guide, but the shape of the population is the real driver.
Is the sample mean of a normal population only approximately normal?
No. For a normal population the sample mean is exactly normal for every n, because linear combinations of independent normal variables are normal. The central limit theorem is needed only for non-normal populations.
Which way does the continuity correction go?
Include or exclude the whole unit bar at the boundary. At least k starts at k − 0.5, at most k ends at k + 0.5, fewer than k ends at k − 0.5, and more than k starts at k + 0.5.
Exam move
Rework two problems side by side, one using an exact normal result and one using the central limit theorem, and write which theorem justified each step. Then drill the four continuity correction cases until the boundary rule is automatic.
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