Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — finding $k$, then a probability
f(x)=kx(4−x) for 0≤x≤4, zero elsewhere. Find k and hence P(1<X<3).
- 1Total area is 1: k∫04(4x−x2)dx=1Integrate only where f=0.
- 2[2x2−3x3]04=32−364=332
- 3k×332=1, so k=323
- 4P(1<X<3)=323[2x2−3x3]13=323(9−35)18−9=9 at x=3; 2−31=35 at x=1.
- 5=323×322=1611=0.6875
Example 2 — expectation and variance by integration
For f(x)=323x(4−x) on 0≤x≤4, find E(X) and Var(X).
- 1E(X)=323∫04(4x2−x3)dxMultiply f by x before integrating.
- 2=323[34x3−4x4]04=323(3256−64)=2Symmetry about x=2 predicts this.
- 3E(X2)=323∫04(4x3−x4)dx=323(256−51024)=524
- 4Var(X)=4.8−22=0.8Subtract the SQUARE of the mean.
Example 3 — cumulative distribution function and median
Find F(x) for f(x)=91x2 on 0≤x≤3, and hence the median.
- 1F(x)=∫0x91t2dt=27x3 for 0≤x≤3
- 2With F(x)=0 for x<0 and F(x)=1 for x>3All three pieces are part of the answer.
- 3Check F(3)=2727=1 ✓Always verify the top of the range.
- 4Median: 27m3=0.5⇒m3=13.5
- 5m=313.5=2.38Inside [0,3] ✓ — and above the midpoint, since the density grows.