Two fair dice are rolled at once. Let X denote the...

Two fair dice are rolled at once. Let $X$ denote the difference in the number of dots that appear on the top faces of the two dice. Thus for example if a one and a five are rolled, $x=4,$ and if two sixes are rolled, $X=0$. a. Construct the probability distribution for $X$. b. Compute the mean $\mu$ of $X$. c. Compute the standard deviation $\sigma$ of $X$.

Two fair dice are rolled at once. Let $X$ denote the difference in the number of dots that
appear on the top faces of the two dice. Thus for example if a one and a five are
rolled, $x=4,$ and if two sixes are rolled, $X=0$.
a. Construct the probability distribution for $X$.
b. Compute the mean $\mu$ of $X$.
c. Compute the standard deviation $\sigma$ of $X$.

Key Concepts

Combinatorial Analysis

Combinatorial analysis involves counting techniques that determine the number of ways outcomes can occur, which is fundamental in calculating probabilities for discrete events. It plays a key role in constructing probability distributions when the outcomes must be enumerated and their frequencies determined from the experimental design.

Expected Value

The expected value, or mean, of a discrete random variable is a measure of its central tendency, calculated as a weighted average of all possible values with their corresponding probabilities. It provides a single summary measure representing the average outcome if the experiment were repeated many times.

Variance and Standard Deviation

Variance measures the spread of the random variable around its mean by averaging the squared differences from the mean, while standard deviation is the square root of the variance, offering a more interpretable measure of dispersion in the same units as the random variable. These concepts are crucial for assessing the degree of variability or uncertainty in outcomes.

Discrete Random Variable

A discrete random variable is a numerical variable that can take on a countable number of distinct values. In probability theory, analyzing such variables involves listing all possible outcomes and their associated probabilities. This concept is central for problems where outcomes, such as differences or counts, are derived from experiments with a finite set of results.

Probability Distribution

The probability distribution of a discrete random variable provides a complete description by assigning a probability to each of its possible values. It is essential for understanding how likely different outcomes are. This distribution is often constructed by counting the number of favorable outcomes for each value and dividing by the total number of outcomes.

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