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The normal distribution

By the end, you can

  • Standardise using the standard deviation
  • Translate probability statements into tail areas
  • Combine independent normal quantities

about 40 minutes for the lesson · practise at your own pace.

The central idea

For X∼N(μ,σ2)X\sim N(\mu,\sigma^2), Z=(X−μ)/σZ=(X-\mu)/\sigma has standard normal distribution. The second parameter is the variance; divide by its square root to standardise. Probabilities are areas under a continuous density, so P(X=x)=0. Linear combinations of independent normal variables remain normal, with variances adding after scaling.

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Worked example

If X∼N(10,4)X\sim N(10,4), its standard deviation is 22. P(X<12)=P(Z<1)≈0.8413P(X<12)=P(Z<1)\approx0.8413. For Y=3X+1Y=3X+1, E(Y)=31E(Y)=31 and Var⁡(Y)=36\operatorname{Var}(Y)=36.

Subtracting independent variables still adds their variances; subtracting the same variable from itself is a different situation.

Check your understanding

A measurement X is modelled as normal, with unknown mean and standard deviation. It has P(X<40)=0.10P(X<40)=0.10 and P(X>65)=0.20P(X>65)=0.20. Estimate both parameters and find the upper cutoff exceeded by 5% of measurements. State the lower-tail probabilities used and verify your fitted parameters against both original statements.

Mastery learning objectives

At the end of this chapter, I should be able to:

Model and compute normal probabilities

Understand the normal distribution as a continuous probability model, including its mean, variance and notation N(μ,σ2)N(\mu,\sigma^2).

Make computations involving the standard normal distribution.

Find P(X<x1)P(X<x_1) or a related probability given x1,μ,σx_1,\mu,\sigma.

Use the symmetry and properties of the normal curve to compute probabilities.

Determine a relationship between x1,μ,σx_1,\mu,\sigma given P(X<x1)P(X<x_1) or a related probability.

Transform and combine variables

Solve problems using E(aX+b)E(aX+b) and Var⁡(aX+b)\operatorname{Var}(aX+b).

Solve problems using E(aX+bY)E(aX+bY) and Var⁡(aX+bY)\operatorname{Var}(aX+bY) where X,YX,Y are independent.

Choose a practice question

The source’s two checkpoints are after lecture and after tutorial. Its full question map is retained below; a blank cell is blank in the source. The two combination bullets share one printed reference cell.

Common question typeLecture examplesDiscussionReview problems (follow up)
Decide whether N(μ,σ2)N(\mu,\sigma^2) is a suitable model23(i)
Find P(X<x1)P(X<x_1) or a related probability given x1,μ,σx_1,\mu,\sigma1, 41, 3(first part), 6, 7(a), 9(first part)1(i), 2(ii), 3(ii), 5(ii), 6(i), 10(i)&(ii)
Find x1x_1 given a probability, μ,σ\mu,\sigma2, 365(i), 7(i), 8(i)
Use standardisation to solve for μ\mu and/or σ\sigma5, 63(second part), 4, 53(iii), 5(i), 8(i), 9(i), 10(iii), 11(i)
Linear combinations of independent normals; sums of independent identically distributed normals7, 8, 9, 107(b)–(e), 8, 9, 10, 11, 12, 131(ii)(iii), 2(i)(iii), 3(iv), 4, 5(first part), 6(iii), 7(iii), 8(ii), 9(ii)(iv)(v)(vi), 10(iv), 11(ii)
Prior topic: Binomial Distribution7(ii), 9(iii)

For independent X,YX,Y, the map gives aX±bY∼N(aμX±bμY,a2σX2+b2σY2)aX\pm bY\sim N(a\mu_X\pm b\mu_Y,a^2\sigma_X^2+b^2\sigma_Y^2). For independent identically distributed observations, X1+⋯+Xn∼N(nμ,nσ2)X_1+\cdots+X_n\sim N(n\mu,n\sigma^2). Retain the chapter’s qualifications for zero variance, dependence and the distinction between a sum and nXnX.

Additional lecture-only Review12: maximise an interval probability is printed after the source map and is not listed among its original routes.

Practice questions

41 examples and practice questions, in 7 sections.

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