Question

Two dice are tossed. Let X be the absolute difference in the number of dots facing up. (a) Find and plot the PMF of X. (b) Find the probability that X lessthanorequalto 2. (c) Find E[X] and Var[X].

Answer 1

To find the PMF of X when two dice are tossed, we need to determine the probability of each possible value of X. The PMF of X is: X = 0: 1/6, X = 1: 5/18, X = 2: 5/18, X = 3: 1/6, X = 4: 1/18, X = 5: 1/18. The probability that X ≤ 2 is 11/18. The expected value (E[X]) is 2.5 and the variance (Var[X]) is 1.9444.

Step-by-step explanation:

a. To find the PMF (Probability Mass Function) of X, we need to determine the probability of each possible value of X. Since there are two dice, each with six sides, there are 36 possible outcomes.
Here is the PMF of X:

X = 0: P(X=0) = 6/36 = 1/6
X = 1: P(X=1) = 10/36 = 5/18
X = 2: P(X=2) = 10/36 = 5/18
X = 3: P(X=3) = 6/36 = 1/6
X = 4: P(X=4) = 2/36 = 1/18
X = 5: P(X=5) = 2/36 = 1/18

b. The probability that X ≤ 2 is equal to the sum of the probabilities of X = 0, 1, and 2.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 1/6 + 5/18 + 5/18 = 11/18

c. To find E[X] (the expected value), we multiply each value of X by its corresponding probability and sum up the results.
E[X] = (0 * 1/6) + (1 * 5/18) + (2 * 5/18) + (3 * 1/6) + (4 * 1/18) + (5 * 1/18) = 2.5

To find Var[X] (the variance), we need to calculate the square of the difference between each value of X and the expected value, multiply it by its corresponding probability, and sum up the results.
Var[X] = ((0 - 2.5)^2 * 1/6) + ((1 - 2.5)^2 * 5/18) + ((2 - 2.5)^2 * 5/18) + ((3 - 2.5)^2 * 1/6) + ((4 - 2.5)^2 * 1/18) + ((5 - 2.5)^2 * 1/18) = 1.9444

Answer 2

To find and plot the PMF of X, we need to determine the probabilities of each possible outcome. To find the probability that X is less than or equal to 2, we need to calculate the sum of the probabilities for X=0, X=1, and X=2. To find the expected value (mean) and variance of X, we use the formulas for discrete random variables.

Step-by-step explanation:

a. Sketch a graph of the probability distribution of X.

The random variable X represents the absolute difference in the number of dots facing up when two dice are tossed. Since each die has 6 sides with numbers from 1 to 6, the possible outcomes for X can range from 0 to 5. The PMF (probability mass function) for X can be represented as follows:

P(X=0) = 1/36

P(X=1) = 2/36

P(X=2) = 4/36

P(X=3) = 6/36

P(X=4) = 8/36

P(X=5) = 10/36

Plotting these probabilities on a graph will show a histogram with X values on the x-axis and corresponding probabilities on the y-axis.

b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X.

To calculate the mean of X, we multiply each X value by its corresponding probability and then sum them up. The mean can be calculated as: (mean) = (0 * 1/36) + (1 * 2/36) + (2 * 4/36) + (3 * 6/36) + (4 * 8/36) + (5 * 10/36) = 2.5

To calculate the standard deviation of X, we need to find the variance first. The variance is calculated as: (variance) = [(0-2.5)^2 * 1/36] + [(1-2.5)^2 * 2/36] + [(2-2.5)^2 * 4/36] + [(3-2.5)^2 * 6/36] + [(4-2.5)^2 * 8/36] + [(5-2.5)^2 * 10/36]. Once we have the variance, the standard deviation is the square root of the variance.