Density normalisation, interval probabilities and expectation.
Example 1
A probability density is f(x)= kx for 0≤x≤3, and zero elsewhere. Find k.
Show worked answer
Answer: 92
- Total density area must equal 1.
- ∫03 kx dx=k29=1.
- Therefore k=92.
Example 2
X has density f(x)=92x on [0,3], and zero elsewhere. Find P(X≤512).
f(x)
012300.51 x
f(x)=92xShow worked answer
Answer: 2516
- Integrate the density from 0 to 512.
- An antiderivative is 9x2.
- At the upper limit this gives 2516.
Example 3
X has density f(x)=362x on [0,6], and zero elsewhere. Find E(X).
f(x)
024600.250.50.75 x
f(x)=362xShow worked answer
Answer: 312
- E(X)=∫xf(x)dx over its support.
- Integrate 362x2 from 0 to 6.
- E(X)=3×362×216=312.