Probability I · Random variables and distribution functions
Two independent attempts have success probabilities 1/5 and 4/5
Problem
Two independent attempts have success probabilities \(1/5\) and \(4/5\). Find the distribution of their success count X and show that it is not binomial with two trials for any common parameter p. Now consider two reports produced by copying the outcome of one fair success-or-failure trial: both reports are successes when that trial succeeds, and both are failures otherwise. Find the success-count distribution Y. Compare both models with two independent fair trials, explaining which binomial assumption fails in each model.
Hint
List the four possible success-and-failure configurations in the first model.
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