5.5 The normal distribution

X~N(μ,σ²) is centred at μ with standard deviation σ. Standardise using Z=(X−μ)/σ, where Z~N(0,1), then use symmetry or normal tables/calculator.

State whether a probability is a lower tail, upper tail or interval. A continuous variable has P(X=a)=0, so endpoint inclusion does not change a probability.

For X~N(10,4), P(X<12)=P(Z<1)=0.8413.

σ is the standard deviation, not the variance; writing N(μ,σ) in a convention that expects variance changes the model.

Use the normal CDF for P(X≤x), complements for upper tails and differences for intervals. Inverse normal questions solve for x from a stated probability.

Draw the tail or interval, standardise consistently and undo the transformation only after finding the z-value. Check whether the resulting value lies in the physical domain.

The central 90% interval for a normal variable uses z≈±1.645, so bounds are μ±1.645σ.

“At least” and “more than” are upper tails, while “between” requires subtracting two cumulative probabilities.

If X counts successes in n independent trials with constant success probability p, X~B(n,p) and P(X=r)=C(n,r)p^r(1−p)^{n−r}.

Check fixed n, two outcomes per trial, independence and constant p. Use complements or cumulative probabilities for “at least” and “at most” wording.

For X~B(10,0.3), P(X=2)=C(10,2)0.3²0.7⁸.

Binomial trials need not be equally likely across outcomes, but p must remain constant; sampling without replacement may break independence.