Probability Distribution of Y:
When two coins are tossed, the possible outcomes for each coin are "head" (H) or "tail" (T). Let's consider the possible outcomes for the two coins together:

- HH (both coins show heads)
- HT (first coin shows heads, second coin shows tails)
- TH (first coin shows tails, second coin shows heads)
- TT (both coins show tails)

The number of "head" occurrences, Y, can take the values 0, 1, or 2 depending on the number of heads obtained.

The probability distribution of Y is as follows:

- P(Y = 0) = P(TT) = 1/4
- P(Y = 1) = P(HT or TH) = 2/4 = 1/2
- P(Y = 2) = P(HH) = 1/4

Cumulative Probability Distribution of Y:
The cumulative probability distribution of Y gives the probability that Y takes on a value less than or equal to a certain value. Let's calculate the cumulative probabilities for each value of Y:

- P(Y ≤ 0) = P(Y = 0) = 1/4
- P(Y ≤ 1) = P(Y = 0) + P(Y = 1) = 1/4 + 1/2 = 3/4
- P(Y ≤ 2) = P(Y = 0) + P(Y = 1) + P(Y = 2) = 1/4 + 1/2 + 1/4 = 1

Mean of Y:
The mean of a probability distribution is the expected value of the random variable. To find the mean of Y, we multiply each possible value of Y by its corresponding probability and sum them up:

- E(Y) = 0 * P(Y = 0) + 1 * P(Y = 1) + 2 * P(Y = 2)
= 0 * 1/4 + 1 * 1/2 + 2 * 1/4
= 0 + 1/2 + 1/2
= 1

Therefore, the mean of Y is 1.

Variance of Y:
The variance of a probability distribution measures the spread of the distribution around the mean. It is calculated using the formula:

- Var(Y) = E((Y - E(Y))^2)

To find the variance of Y, we substitute the values of Y and its corresponding probabilities into the formula:

- Var(Y) = (0 - 1)^2 * P(Y = 0) + (1 - 1)^2 * P(Y = 1) + (2 - 1)^2 * P(Y = 2)
= 1^2 * 1/4 + 0^2 * 1/2 + 1^2 * 1/4
= 1/4 + 0 + 1/4
= 1/2

Therefore, the variance of Y is 1/2.

Summary:
- The probability distribution of Y for two coin tosses is given by P(Y = 0) = 1/4, P(Y = 1) = 1/