Problem 3 - 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) \(E(X)\). $$ \begin{array}{l|lllll} x_{i} & -2 & -1 & 0 & 1 & 2 \\ \hline p_{i} & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \end{array} $$

Step-by-Step Solution

Answer

\(P(X \geq 2) = 0.2\); \(E(X) = 0\).

Step 1: Understanding the Problem

We have a discrete random variable \(X\) with a given probability distribution. Each value \(x_i\) has an equal probability \(p_i = 0.2\). We need to calculate \(P(X \geq 2)\) and the expected value \(E(X)\).

Step 2: Calculating \(P(X \geq 2)\)

To find \(P(X \geq 2)\), we focus on the values of \(X\) that satisfy this inequality. From the table, the only value is \(x = 2\), with probability \(p = 0.2\). Therefore, \(P(X \geq 2) = 0.2\).

Step 3: Using the Formula for Expected Value

The formula for the expected value \(E(X)\) of a discrete random variable is given by \(E(X) = \sum_{i} x_i p_i\). Here we will sum the products of each \(x_i\) and \(p_i\).

Step 4: Calculating Each Term for \(E(X)\)

Calculate each term: \(-2 \times 0.2 = -0.4\), \(-1 \times 0.2 = -0.2\), \(0 \times 0.2 = 0\), \(1 \times 0.2 = 0.2\), and \(2 \times 0.2 = 0.4\).

Step 5: Summing the Results for \(E(X)\)

Sum the results: \(-0.4 + (-0.2) + 0 + 0.2 + 0.4 = 0\). Thus, \(E(X) = 0\).

Key Concepts

Discrete Random Variable

A discrete random variable is a type of random variable that takes on a countable number of distinct values. In simpler terms, you can list these values as individual, separate items.
These variables often represent things we can count, such as the number of students in a class, or in the context of our problem, the different possible outcomes of a random experiment.
In our exercise, the random variable \(X\) can take on the values of \(-2, -1, 0, 1,\) and \(2\). Let's think of it like drawing marbles from a bag, each number representing a different colored marble.
A discrete random variable is fundamental in probability, providing the backbone for concepts like probability distributions and expected value.

Expected Value

The expected value, sometimes called the mean, is a key concept in probability that helps us understand the center or average of a random variable.
It gives us an idea of what to expect on "average" if we were to repeat an experiment many times.

• In the case of our exercise, the expected value of \(X\), denoted as \(E(X)\), involves multiplying each possible outcome \(x_i\) by its corresponding probability \(p_i\) and then summing all these products up.

For example, if you toss a fair six-sided die, the expected value is \(3.5\), because:
\[E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5\]
In the solved problem, each outcome was equally likely, making the sum precisely zero due to the symmetric nature of the distribution around zero.

Probability

Probability is the measure of the likelihood that an event will occur, expressed as a number between 0 and 1.
In our exercise, each outcome of the random variable was equally likely, each having a probability of \(0.2\).

• A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event.
• The sum of probabilities for all possible outcomes for a discrete random variable must equal 1.

To answer parts of our problem, we calculated \(P(X \geq 2)\), which focuses on the probability of event \(X\) being 2 or higher. Only one outcome, when \(X = 2\), fits this criterion with its probability of \(0.2\). Basic probability like this forms the bedrock of more complex statistical theories.

Random Variable

A random variable is a fundamental concept in statistics and probability, used to represent numerical outcomes of random phenomena.
These variables can be discrete, like in our exercise, or continuous, meaning they could take on any value within a range.

• A random variable acts as a function from a sample space (the set of all possible outcomes) to the real numbers, assigning a numerical outcome to each scenario.
• For example, think of rolling a die. Each face corresponds to a number from 1 to 6, making it a random variable.

In our exercise, \(X\) is a random variable representing the possible outcomes \(-2, -1, 0, 1,\) and \(2\). Random variables are the cornerstone of predicting and modeling real-world phenomena, helping us understand and compute probabilities and expectations.