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Random Variables and Distribution Practice

Allow 60–90 minutes. Derive support before choosing a distribution. Use the worked examples for B1, then attempt the rest unaided; solutions are separate.

  1. B1 — finite pool. Six machines include two defects. Choose two uniformly without replacement. Derive the PMF, mean, and variance of defect count. Compare with Binomial(2,1/3), including variance.
  2. B2 — stopped attempts. Each attempt independently fails with probability .2. Stop on success, with at most four total attempts. List terminal paths and derive expected attempts, failures, backoffs, and success probability. At50 ms per backoff, find expected added waiting.
  3. B3 — a density. A density is f(x)=c*x on [0,2], zero elsewhere. Determine c, P(X<=1), E[X], and the median. Explain why density height is not a point probability.
  4. B4 — a process model. A homogeneous Poisson process has rate8 arrivals/hour. What is the count parameter for15 minutes? Find the probability of zero arrivals and mean waiting time in minutes. State the assumptions, then describe one bursty process that violates them.
  5. B5 — a transformation. X equals -1,0,2 with probabilities1/4,1/2,1/4. Find the PMF of X², E[X], E[X²], and Var(X). Does squaring commute with expectation here?

Repair by enumerating small outcomes, normalizing the PMF/density, and checking units. Do not select a distribution only because the output is a count: the sampling and stopping mechanisms matter.