1. Normalising a discrete distribution
Difficulty: ★☆☆A random variable \(X\) takes only the values \(-2,1,4\), with probabilities \(0.25,k,0.15\), respectively. What is \(k\)?
Hint
2. Discrete inequalities
Difficulty: ★☆☆The probability distribution of \(X\) is shown. What is \(P(1<X\le4)\)?
| \(x\) | \(P(X=x)\) |
|---|---|
| \(0\) | \(0.10\) |
| \(1\) | \(0.25\) |
| \(2\) | \(0.35\) |
| \(4\) | \(0.30\) |
Hint
3. Expected value
Difficulty: ★☆☆A game score \(X\) is \(-3,2,5\) with probabilities \(0.2,0.5,0.3\), respectively. What is \(\mathrm{E}(X)\)?
Hint
4. Variance from two moments
Difficulty: ★★☆\(X\) takes the values \(1,3,5\) with probabilities \(1/4,1/2,1/4\), respectively. What is \(\operatorname{Var}(X)\)?
Hint
5. Expectation of a nonlinear function
Difficulty: ★★☆\(X\) takes the values \(-1,0,2\) with probabilities \(1/4,1/2,1/4\), respectively. What is \(\mathrm{E}[(X+2)^2]\)?
Hint
6. Fair entry fee
Difficulty: ★★☆A game pays a gross prize of \(0\), \(4\) or \(20\) pounds with probabilities \(0.70\), \(0.25\) and \(0.05\), respectively. A fixed entry fee is charged for every play. What entry fee makes the expected net gain zero?
Hint
7. Discrete uniform variance
Difficulty: ★★☆\(X\) is uniformly distributed on \(\{-4,-1,2,5\}\). What is \(\operatorname{Var}(X)\)?
Hint
8. Finding a probability from the mean
Difficulty: ★★☆\(X\) takes values \(0,2,5\) with probabilities \(p,0.4,0.6-p\), respectively. Given that \(\mathrm{E}(X)=2.3\), find \(p\).
Hint
9. Recovering probabilities from moments
Difficulty: ★★★\(X\) takes only the values \(0,1,3\). It is known that \(\mathrm{E}(X)=3/2\) and \(\mathrm{E}(X^2)=7/2\). What is \(P(X=0)\)?
Hint
10. Constructing a new discrete variable
Difficulty: ★★★A spinner gives score \(1\) with probability \(1/4\) and score \(3\) with probability \(3/4\). It is spun twice independently. Let \(M\) be the larger of the two scores. What is \(\mathrm{E}(M)\)?