The CDF of X is
P(X ≤ 1) = 1/36
P(X ≤ 2) = 1/18
P(X ≤ 3) = 1/12
P(X ≤ 4) = 1/9
P(X ≤ 5) = 5/18
P(X ≤ 6) = 1/6
Yes, X is a random variable, as it takes on different values with different probabilities. The cumulative distribution function (CDF) of X represents the probability that X is less than or equal to a specific value.
To determine the CDF, consider the possible outcomes and their probabilities:
- X = 1: Occurs only when both dice show 1, with a probability of (1/6) * (1/6) = 1/36.
- X = 2: Occurs in two ways - (1, 2) and (2, 1) - both with a probability of (1/6) * (1/6) = 1/36. So, P(X ≤ 2) = 2/36 = 1/18.
- X = 3: Occurs in three ways - (1, 3), (3, 1), and (2, 2) - each with a probability of (1/6) * (1/6) = 1/36. So, P(X ≤ 3) = 3/36 = 1/12.
- X = 4: Occurs in four ways - (1, 4), (4, 1), (2, 3), and (3, 2) - each with a probability of (1/6) * (1/6) = 1/36. So, P(X ≤ 4) = 4/36 = 1/9.
- X = 5: Occurs in five ways - (1, 5), (5, 1), (2, 4), (4, 2), and (3, 3) - each with a probability of (1/6) * (1/6) = 1/36. So, P(X ≤ 5) = 5/36 = 5/18.
- X = 6: Occurs in six ways - (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), and (4, 3) - each with a probability of (1/6) * (1/6) = 1/36. So, P(X ≤ 6) = 6/36 = 1/6.
The CDF of X can be summarized as:
P(X ≤ 1) = 1/36
P(X ≤ 2) = 1/18
P(X ≤ 3) = 1/12
P(X ≤ 4) = 1/9
P(X ≤ 5) = 5/18
P(X ≤ 6) = 1/6