Question

two dice are thrown simultaneously. let x be the maximum of the numbers that the two dice show. is this x a random variable? if yes find the cumulative distribution function of x.

Answer 1

The maximum value of the two dice, X, is a discrete random variable, and its CDF can be calculated by considering the probability P(X ≤ x) for each possible outcome x (1 through 6). By counting the combinations for each threshold and dividing by the total number of outcomes (36), we construct the CDF.

Step-by-step explanation:

Yes, the maximum value that appears on either of the two dice, denoted as X, is a random variable because it takes on numerical outcomes from the experiment of rolling two dice. In this case, X is a discrete random variable because it can take on a finite number of possible values.

To find the cumulative distribution function (CDF) of X, we need to calculate the probability that X is less than or equal to a value x, which is P(X ≤ x). We'll do this for each possible value of X (which are 1, 2, 3, 4, 5, 6 for a six-sided die).

1. For P(X ≤ 1), since the maximum value of a die roll cannot be less than 1, this is just the probability of rolling a pair of ones: 1/36.
2. For P(X ≤ 2), we include the probabilities of rolling a maximum of 1 or 2. This includes 1-1, 2-1, 1-2, and 2-2: (1+2+2+1)/36 = 1/9.
3. This pattern continues for values of x from 3 to 6, where for each value of x we count the combinations with at least one die showing x and the other showing at most x.

The CDF of X increases as the value of x increases, giving us the cumulative probability for each value X could take.