Question

A fair coin is tossed 4 times. Let x denote the number of times heads occur and y denote the length of the longest string. Find the covariance (cov) between x and y.

Answer 1

To find the covariance between x (number of heads) and y (length of longest string) in 4 coin tosses, we calculate the joint probability distribution. Using the formula cov(x, y) = E(xy) - E(x)E(y), we can find the covariance by calculating the expected values of xy, x, and y.

Step-by-step explanation:

To find the covariance between x (the number of times heads occur) and y (the length of the longest string), we need to calculate the joint probability distribution of x and y. Let's start by finding the probability of each possible outcome for x and y.

Since the coin is fair, the probability of getting a head on each toss is 0.5. The number of heads in 4 tosses can range from 0 to 4, so we have x = 0, 1, 2, 3, 4. The length of the longest string can range from 0 to 4 as well, so we have y = 0, 1, 2, 3, 4.

To find the joint probability of each outcome, we multiply the probabilities of getting x heads and y as the longest string. For example, the probability of getting 0 heads and a longest string of 0 would be (0.5)^0 * (0.5)^0 = 1 * 1 = 1.

By calculating the joint probabilities for all possible outcomes, we can construct a table and then find the covariance using the formula:

cov(x, y) = E(xy) - E(x)E(y)