Probability I · Expectation, moments, and transforms
A random variable X takes values 0, 1, and 3 with probabilities 1/2,1/3,1/6, respectively
Problem
A random variable \(X\) takes values 0, 1, and 3 with probabilities \(1/2,1/3,1/6\), respectively. Calculate \(\mathbb E X\) and \(\mathbb E[(X-1)^2]\) directly from this distribution. Compare the latter with \((\mathbb E X-1)^2\), and explain why a nonlinear function cannot generally be moved outside an expectation.
Hint
Make a row for each possible value of \(X\) and retain its probability.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Probability I sample problem: Try the free sample problem.
More Probability I practice problems
- A spinner has four equally likely sectors, labelled A₁,A₂,B,CProbability spaces, axioms, and rigorous counting
- Choose a three-element subset uniformly from {1,…,9}, and let M be its largest elementProbability spaces, axioms, and rigorous counting
- Roll two fair six-sided dice independently and record the ordered pairConditional probability and independence
- Independent rounds of a game have three possible results: win with probability 2/7…Conditional probability and independence
- Roll two fair six-sided dice independently and let X be the absolute difference between…Random variables and distribution functions
- Two independent attempts have success probabilities 1/5 and 4/5Random variables and distribution functions
- A sorting device independently sends each of six incoming pieces to tray A, B, or C…Joint distributions and conditional expectation
- Conditioning changes the interval on a triangular supportJoint distributions and conditional expectation
- A weighted triangle is not uniform on its conditional slicesJoint distributions and conditional expectation