Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find \((a)\) \(P(X \geq 2)\) and \((b) E(X)\). $$ \begin{array}{l|lll} x_{i} & -0.1 & 100 & 1000 \\ \hline p_{i} & 0.980 & 0.018 & 0.002 \end{array} $$

Step-by-Step Solution

Answer

(a) 0, (b) 3.702

Step 1: Identify Probabilities for Part (a)

We are asked to find \(P(X \geq 2)\). Looking at the possible outcomes for \(X\), which are \(-0.1, 100,\) and \(1000\), there are no outcomes equal to or greater than 2. Therefore, the probability \(P(X \geq 2) = 0\).

Step 2: Calculate Expected Value for Part (b)

To find the expected value \(E(X)\), use the formula \(E(X) = \sum_{i} x_{i} p_{i}\). Substitute the values from the distribution: \(E(X) = (-0.1 \times 0.980) + (100 \times 0.018) + (1000 \times 0.002)\).

Step 3: Perform Calculations for Expected Value

Calculate each term and add them: \((-0.1 \times 0.980) = -0.098\), \((100 \times 0.018) = 1.8\), and \((1000 \times 0.002) = 2\). Add these results: \(-0.098 + 1.8 + 2 = 3.702\). Thus, \(E(X) = 3.702\).

Key Concepts

Understanding Expected Value

The expected value is a fundamental concept in probability theory. It gives you an idea of what to anticipate as an 'average' result if you were to repeat an experiment many times.

In simpler terms, it represents the long-term average of a random variable. For a discrete random variable, the expected value is calculated using its probability distribution. The formula is as follows:

\[ E(X) = \sum_{i} x_{i} p_{i} \]

Here, \(x_i\) are the possible values that the random variable \(X\) can take, and \(p_i\) denotes the probability associated with each \(x_i\). By multiplying each outcome by its probability and summing up these products, you find the expected value.

Remember, the expected value does not need to be an outcome that is possible with a single occurrence of the random variable. Instead, it serves as a statistical mean that reflects the distribution of probabilities around the possible outcomes.

Exploring Random Variables

A random variable is a mathematical function that maps outcomes of a random process to numbers. It is crucial in the study of probability theory because it quantifies outcomes.

There are two main types of random variables:

• **Discrete Random Variables:** These take on countable values, typically finite or infinite sequences like \(-0.1, 100, 1000\) in our example. They are described using discrete probability distributions.
• **Continuous Random Variables:** These take on values from a continuous range, like the exact amount of rain on a particular day.

In our exercise, \(X\) is a discrete random variable, representing possible outcomes of an event with specific probabilities. Discrete random variables are often easier to handle because the calculations involve simple sums and products, as shown in the expected value calculation.

Basics of Probability Theory

Probability theory is the mathematical framework that underpins the study of random events and phenomena. It's a way to measure uncertainty and assign numerical values to the likelihood of different occurrences.

At its core, probability theory involves determining the probability of potential outcomes. These outcomes constitute a sample space, where each event's probability must lie between 0 and 1. The total of all probabilities in a probability distribution equals 1 because one of the possible outcomes must occur.

Understanding probability theory and its concepts, such as the expected value and random variables, empowers you to interpret and predict real-world scenarios scientifically. This foundational knowledge is essential in diverse fields such as finance, environmental studies, and engineering, where making informed predictions based on data is crucial.