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Calculate probabilities for normal distributions

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Use the combined distribution first

Once W has a normal distribution with positive standard deviation, standardise a boundary by subtracting the mean and dividing by the standard deviation. Preserve the event direction; the positive divisor does not reverse an inequality

Year 12, VCE Units 3 & 4, Chapter 8, 1 difficulty level, Calculator allowedBuild a worksheet

Linear combinations

Keep variable identity, coefficient signs and joint assumptions explicit

  1. Mean The mean of a constant plus a weighted sum is the constant plus the sum of each weight times its variable’s mean, regardless of independence
  2. Independent variance For independent variables, the variance of a constant plus a weighted sum is the sum of each squared weight times its variable’s variance. The standard deviation is the positive square root
  3. Normal inputs Independent normals give a normal combination when its variance is positive
  4. Repeated source Collect coefficients of the same variable before using independence
  5. Probability Standardise each boundary using the combined mean and standard deviation; round at the end

Worked example

Independent normal variables X and Y have means 50 and 30 and variances 9 and 16, respectively. Find the probability that X exceeds Y by more than 25

\[\Pr(X-Y>25)\]
  1. \(\displaystyle X-Y\sim N(20,25)\)Subtract the means but add the independent variances
  2. \(\displaystyle z=\frac{25-20}{5}=1\)Use the combined standard deviation of 5
  3. \(\displaystyle \Pr(Z>1)=1-\Phi(1)\approx0.1587\)Take the upper tail and round to four decimal places

Common mistakes

Difference variance
✕\(\displaystyle \operatorname{Var}(X-Y)=\operatorname{Var}(X)-\operatorname{Var}(Y)\quad\text{for independent variables}\)✓\(\displaystyle \operatorname{Var}(X-Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)\)
The coefficient -1 is squared
Same draw twice
✕\(\displaystyle \operatorname{Var}(2X-X)=5\operatorname{Var}(X)\)✓\(\displaystyle \operatorname{Var}(2X-X)=\operatorname{Var}(X)\)
The two X terms are the same random quantity; collect them first
Normal parameters
✕Divide the centred boundary by the variance✓Divide by the positive standard deviation
N(μ,σ²) states the variance as its second parameter

Difficulty levels

examStandardise a normal boundary only

Questions about this topic

Must the inputs be identically distributed?+

No. Keep each variable’s own mean and variance

Is an independent sum always normal?+

No. The exact normal combination theorem used here assumes independent normal inputs

What if both coefficients are zero?+

The result is constant. Its variance is zero and ordinary standardisation is undefined

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 8: Probability and statistical inference

In this chapter

Standardise tails of three independent normal variablesCombine independent random variables and normal distributionsConstruct finite independent sums and compare repeated copies with scalingCalculate probabilities for normal distributions8B Distribution of sample means: Use normal sampling distributions and standard errorsConstruct sample means and interpret finite sampling distributionsConstruct and interpret confidence intervals for a population mean8C Confidence intervals for population means: Use normal sampling distributions and standard errorsPredict interval coverage and plan a width reductionForm and interpret hypothesis tests for a population mean8D Hypothesis tests for population means: Use normal sampling distributions and standard errorsCalculate theoretical test errors and supplied-tail boundaries