Suppose we flip two 'fair' coins, and the random variables are defined as: X1 = [ 0 if tail, and 1 if heads ]X2 = [ 0 if tail, and 1 if heads ]Y = min [X1, X2] E(Y) =
Question
Suppose we flip two 'fair' coins, and the random variables are defined as: X1 = [ 0 if tail, and 1 if heads ]X2 = [ 0 if tail, and 1 if heads ]Y = min [X1, X2] E(Y) =
Solution
The text is in English. Here is the output:
To find the expected value E(Y), we need to consider all possible outcomes of flipping two fair coins and the corresponding values of Y, which is defined as the minimum of X1 and X2.
- If both coins show tails, then X1 = 0 and X2 = 0. So, Y = min[0, 0] = 0.
- If one coin shows heads and the other shows tails, then X1 = 1 and X2 = 0 (or vice versa). So, Y = min[1, 0] = 0.
- If both coins show heads, then X1 = 1 and X2 = 1. So, Y = min[1, 1] = 1.
The probability of each outcome is as follows:
- The probability of both coins showing tails is (1/2) * (1/2) = 1/4.
- The probability of one coin showing heads and the other showing tails is 2 * (1/2) * (1/2) = 1/2. (The factor of 2 is because the head can appear on either the first or the second coin.)
- The probability of both coins showing heads is (1/2) * (1/2) = 1/4.
Therefore, the expected value E(Y) is given by the sum of the product of each outcome and its probability:
E(Y) = 0 * (1/4) + 0 * (1/2) + 1 * (1/4) = 1/4.
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