Problem 3 - Calculus for Biology and Medicine

Question

Roll a fair die twice. Let \(X\) be the random variable that gives the absolute value of the differences betwen the two numbers. Find the probability mass function describing the distribution of \(X\).

Step-by-Step Solution

Answer

The PMF of \(X\) is: \(P(X = 0) = \frac{1}{6}\), \(P(X = 1) = \frac{5}{18}\), \(P(X = 2) = \frac{2}{9}\), \(P(X = 3) = \frac{1}{6}\), \(P(X = 4) = \frac{1}{9}\), \(P(X = 5) = \frac{1}{18}\).

Step 1: Determine Possible Outcomes

When we roll a die twice, each roll has 6 possible outcomes: 1, 2, 3, 4, 5, or 6. So, there are a total of 36 possible outcomes for two rolls (6 outcomes for the first roll multiplied by 6 outcomes for the second roll): \[(1,1), (1,2), \ (1,3), \ldots, (6,6)\]

Step 2: Define Random Variable X

The random variable \(X\) is defined as the absolute value of the difference between the numbers rolled: \[X = |Y_1 - Y_2|\] where \(Y_1\) is the result of the first roll and \(Y_2\) is the result of the second roll. \(X\) can take values in the set \{0, 1, 2, 3, 4, 5\}.

Step 3: Count Outcomes for Each X Value

Calculate the number of outcomes that lead to each possible value of \(X\):- \(X = 0\): \((1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\) — 6 outcomes- \(X = 1\): \((1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5)\) — 10 outcomes- \(X = 2\): \((1,3), (3,1), (2,4), (4,2), (3,5), (5,3), (4,6), (6,4)\) — 8 outcomes- \(X = 3\): \((1,4), (4,1), (2,5), (5,2), (3,6), (6,3)\) — 6 outcomes- \(X = 4\): \((1,5), (5,1), (2,6), (6,2)\) — 4 outcomes- \(X = 5\): \((1,6), (6,1)\) — 2 outcomes.

Step 4: Calculate Probabilities

The probability mass function (PMF) of \(X\) is calculated by dividing the number of successful outcomes for each \(X\) value by the total number of outcomes (36):- \(P(X = 0) = \frac{6}{36} = \frac{1}{6}\)- \(P(X = 1) = \frac{10}{36} = \frac{5}{18}\)- \(P(X = 2) = \frac{8}{36} = \frac{2}{9}\)- \(P(X = 3) = \frac{6}{36} = \frac{1}{6}\)- \(P(X = 4) = \frac{4}{36} = \frac{1}{9}\)- \(P(X = 5) = \frac{2}{36} = \frac{1}{18}\).

Key Concepts

Random Variables

A random variable is a fundamental concept in probability and statistics. It is essentially a numerical description of the outcomes of a random event. Here, it serves as a bridge between real-world contexts and mathematical frameworks. For instance, when rolling a die twice:

• Each roll produces outcomes that can be captured in a numerical form.
• These outcomes when combined (for example, (1,2) or (4,5)) form the basis of our random variable.
• In our specific problem, the random variable, denoted as \(X\), is used to find the absolute difference between die rolls.

The possible values that \(X\) can assume are 0 through 5, representing the absolute differences between each pair of outcomes. Understanding random variables helps in determining how likely specific events are and creating a smooth transition into more advanced statistical concepts.

Probability Distribution

Probability distribution is a concept that describes how the probabilities are distributed over the various possible outcomes of the random variable. It forms a backbone for predicting outcomes based on historical data. In our problem:

• The probability distribution is captured by the probability mass function (PMF) of \(X\).
• This shows the probability of each possible value of \(X\) occurring.
• The PMF is derived by dividing the number of favorable outcomes by the total number of outcomes when rolling the dice twice.

For example, the probability \(P(X = 0) = \frac{1}{6}\) indicates there is a \(1/6\) chance of the difference between the rolls being 0. Familiarity with probability distributions is crucial, as they provide insights into the behavior of random variables and help determine expected values.

Absolute Difference

An absolute difference is a simple yet powerful concept critical in understanding the variability between outcomes. It measures how much two values differ from each other without considering the direction of difference (i.e., whether one is larger or smaller). For our dice rolling example:

• The absolute difference uses only the magnitude of difference, which can be calculated by \(|Y_1 - Y_2|\).
• It provides straightforward and unbiased measures, smoothing out the effect of any negative outcomes.
• This simplification makes it easier to understand the relationship between values and enables easier calculation of probabilities.

The role of absolute difference can be seen when we compute \(X\) values out of rolling a fair die twice and helps in ensuring the probabilities are equally spread, leading to correct probability distributions.