Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The probability distribution of the random variable \(X\) is shown in the accompanying table: $$\begin{array}{lccccccc}\hline \boldsymbol{x} & -10 & -5 & 0 & 5 & 10 & 15 & 20 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .20 & .15 & .05 & .1 & .25 & .1 & .15 \\ \hline\end{array}$$ Find a. \(P(X=-10)\) b. \(P(X \geq 5)\) c. \(P(-5 \leq X \leq 5)\) d. \(P(X \leq 20)\)

Step-by-Step Solution

Answer

a. \(P(X = -10) = 0.20\) b. \(P(X \geq 5) = 0.60\) c. \(P(-5 \leq X \leq 5) = 0.30\) d. \(P(X \leq 20) = 1.00\)

Step 1: a. Finding P(X = -10)

Based on the probability distribution table, we can directly read the probability P(X = -10). In this case, it is given as 0.20. So, P(X = -10) = 0.20.

Step 2: b. Finding P(X ≥ 5)

To find P(X ≥ 5), we need to sum the probabilities of X = 5, 10, 15, and 20 from the probability distribution table. P(X ≥ 5) = P(X=5) + P(X=10) + P(X=15) + P(X=20) = 0.1 + 0.25 + 0.1 + 0.15 = 0.60.

Step 3: c. Finding P(-5 ≤ X ≤ 5)

To find P(-5 ≤ X ≤ 5), we need to sum the probabilities of X = -5, 0, and 5 from the probability distribution table. P(-5 ≤ X ≤ 5) = P(X=-5) + P(X=0) + P(X=5) = 0.15 + 0.05 + 0.1 = 0.30.

Step 4: d. Finding P(X ≤ 20)

Since we are given the probability distribution of X and all values of X are less or equal to 20, P(X ≤ 20) is equal to the sum of all probabilities in the table. P(X ≤ 20) = P(X=-10) + P(X=-5) + P(X=0) + P(X=5) + P(X=10) + P(X=15) + P(X=20) = 0.20 + 0.15 + 0.05 + 0.1 + 0.25 + 0.1 + 0.15 = 1.00.

Key Concepts

Random Variables

In probability and statistics, a random variable is a variable that can take on different values based on the outcomes of a random phenomenon. It serves as a function to map the outcomes of a random process to real numbers. In essence, random variables help us study and analyze statistical phenomena in a structured manner. For example, when we roll a dice, the outcome can be any number from 1 to 6, so the random variable is the outcome of the dice roll.

Random variables can be classified into two types:

• Discrete random variables: These take on a countable number of distinct values. An example would be the number of boys in a family of three children.
• Continuous random variables: These can take on values from a continuum. An example would be the time it takes for a computer to process a certain command.

In the context of the original exercise, the random variable \(X\) takes on values like -10, -5, 0, 5, 10, 15, and 20. This makes \(X\) a discrete random variable because it only assumes distinct, separate values from a given set.

Probability Calculation

Calculating probabilities with random variables involves understanding the function or mapping of possible outcomes to their respective probabilities. In the given example, we are provided with a probability distribution table that clearly shows the probability that the random variable \(X\) takes on each specific value.

To calculate probabilities for specific ranges or conditions, you simply sum up the probabilities that meet these criteria. For example:

• For calculating \(P(X \geq 5)\), we add the probabilities of \(X\) being 5, 10, 15, and 20. This ends up being \(0.1 + 0.25 + 0.1 + 0.15 = 0.60\).
• To find \(P(-5 \leq X \leq 5)\), we add the probabilities of \(X\) being -5, 0, and 5, resulting in \(0.15 + 0.05 + 0.1 = 0.30\).

Having the probability distribution readily available simplifies the probability calculation process as you can quickly identify and sum the probabilities.

Discrete Probability

Discrete probability involves calculating the likelihood of specific, countable outcomes. This concept is pivotal in scenarios where outcomes are distinct and separate, such as rolling a die or flipping a coin. Here, only a discrete set of outcomes is possible, and each has an associated probability.

The probabilistic nature of the outcomes in our exercise table makes it a discrete probability model, as the values the random variable \(X\) can assume are distinct and finite. Each outcome (or value) of \(X\) from the table comes with a specific probability, demonstrating the essence of discrete probability.

• The sum of these probabilities across all possible outcomes always equals 1, evidencing the exhaustive nature of the outcomes. For instance, for \(X\), when summed, all probabilities add to 1: \(0.20 + 0.15 + 0.05 + 0.1 + 0.25 + 0.1 + 0.15 = 1.00\).
• Each potential outcome, such as \(X = -10\) or \(X = 20\), holds a specific probability, facilitating precise probability calculation for discrete events.

Discrete probability models are a fundamental component of probability theory and are used extensively in statistical analysis to predict outcomes based on specific scenarios.