1. Normalising a constant density
Difficulty: ★☆☆A continuous random variable \(X\) has density \(f(x)=k\) for \(2<x<6\), and \(f(x)=0\) elsewhere. What is \(k\)?
Hint
2. Point probabilities
Difficulty: ★☆☆The density of a continuous random variable is \(f(x)=3x^2/8\) for \(0<x<2\), and zero elsewhere. What is \(\operatorname{P}(X=1)\)?
Hint
3. Area under a decreasing density
Difficulty: ★☆☆A random variable \(X\) has density \(f(x)=2(3-x)/9\) for \(0<x<3\), and zero elsewhere. Find \(\operatorname{P}(X<1)\).
Hint
4. Mean of a continuous distribution
Difficulty: ★★☆The density of \(X\) is \(f(x)=\frac{3}{16}(4-x^2)\) for \(0<x<2\), and zero elsewhere. What is \(\mathrm{E}(X)\)?
Hint
5. Variance from continuous moments
Difficulty: ★★☆A random variable has density \(f(x)=4x^3\) for \(0<x<1\), and zero elsewhere. Find \(\operatorname{Var}(X)\).
Hint
6. Median from an area equation
Difficulty: ★★☆A random variable \(X\) has density \(f(x)=(x+1)/6\) for \(1<x<3\), and zero elsewhere. What is its median?
Hint
7. Converting an event into an interval
Difficulty: ★★☆The density of \(X\) is \(1/6\) for \(-2<X<4\), and zero elsewhere. Find \(\operatorname{P}(X^2<4)\).
Hint
8. Percentile with an infinite support
Difficulty: ★★☆A nonnegative random variable has density \(f(x)=2/(x+1)^3\) for \(x\geq0\), and zero elsewhere. Find its 80th percentile.
Hint
9. Conditional probability from a density
Difficulty: ★★★The density of \(X\) is \(6x(1-x)\) for \(0<x<1\), and zero elsewhere. Given that \(X>1/4\), what is the probability that \(X>1/2\)?
Hint
10. Recovering a density from its mean
Difficulty: ★★★A density is \(f(x)=kx\) for \(0<x<a\), and zero elsewhere, where \(a>0\) and \(k>0\). The mean is \(4\). Which pair \((a,k)\) is correct?