Question

Suppose there are two urns. Urn I contains 100 chips: 30 are labelled 1, 40 are labelled 2, and 30 are labelled 3. Urn 2 contains 100 chips: 20 are labelled 1, Chapter 3: Expectation 183 50 are labelled 2, and 30 are labelled 3. A coin is tossed and if a head is observed, then a chip is randomly drawn from urn 1, otherwise a chip is randomly drawn from urn 2. The value Y on the chip is recorded. If an occurrence of a head on the coin is denoted by X = 1, a tail by X = 0, and X ∼ Bernoulli(3/4), then determine E(X | Y), E(Y | X), E(Y), and E(X)

Answer 1

The given probabilities are

The given conditional probabilities are

Now we deduce the following probabilities.

The conditional probabilities,

The expected values,

Using the property of conditional expectation,