The two continuous distributions that are not the normal — and they come up constantly. Both are solved by finding areas of simple shapes, not by a calculator function, which is exactly why they are examined.
Spot which one from the wording
“Most likely”, “peaks at”, a mode given with a minimum and maximum → triangular.
“Equally likely”, “uniform”, “any value between” with just a minimum and maximum → rectangular (uniform).
Both are continuous, so total area under the graph = 1. Every probability is an area.
Rectangular (uniform) on [a, b]
Height h=\frac{1}{b-a} Because area = width × height = 1. Probability P(c<X<d)=\frac{d-c}{b-a} Just the fraction of the interval you want. Mean \mu=\frac{a+b}{2}
Triangular on [a, b] with mode c
Peak height h=\frac{2}{b-a} Area of a triangle = ½ × base × height = 1, so h = 2/(b−a). Mean \mu=\frac{a+b+c}{3} The average of the three corners. Left-hand area P(X<x)=\frac{(x-a)^2}{(b-a)(c-a)} Valid for a ≤ x ≤ c. Right-hand area P(X>x)=\frac{(b-x)^2}{(b-a)(b-c)} Valid for c ≤ x ≤ b. ✓
You can always fall back on similar triangles
If you cannot recall the formula, draw it. The peak height is 2/(b − a). For any x on the rising side, the small triangle from a to x is similar to the whole left triangle, so its height scales as (x − a)/(c − a) — then the area is just ½ × base × height. Working it out from the picture earns the same marks as quoting the formula, and it is much harder to misremember. Worked example Triangular probability Delivery times are triangular with a minimum of 10 min, a maximum of 40 min, and a most likely time of 20 min. Find P(X < 25). a = 10, b = 40, c = 20. Peak height h = 2/(40 − 10) = 1/15. 25 is on the falling side (between c = 20 and b = 40), so it is easier to find P(X > 25) and subtract. P(X > 25) = (b − x)² / [(b − a)(b − c)] = (40 − 25)² / [(40 − 10)(40 − 20)] = 225 / 600 = 0.375. So P(X < 25) = 1 − 0.375 = 0.625. ✓ P(X < 25) = 0.625. Check it is sensible: 25 is past the mode, so more than half the area should be below it — and 0.625 > 0.5. ✓ ✕
Which formula for which side
The two triangular area formulas are not interchangeable. Use (x − a)²/[(b−a)(c−a)] only when x is on the rising side (a ≤ x ≤ c), and (b − x)²/[(b−a)(b−c)] only when x is on the falling side (c ≤ x ≤ b). Sketch the triangle and mark x first — it takes five seconds and stops you using the wrong one. If x lands past the mode, find the tail and subtract from 1.