Problem 2 - 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) \(E(X)\). $$ \begin{array}{l|lllll} x_{i} & 0 & 1 & 2 & 3 & 4 \\ \hline p_{i} & 0.70 & 0.15 & 0.05 & 0.05 & 0.05 \end{array} $$

Step-by-Step Solution

Answer

(a) 0.15, (b) 0.60.

Step 1: Understand the Problem

We are given a probability distribution for the discrete random variable \(X\) with outcomes 0, 1, 2, 3, and 4, along with their respective probabilities. We need to calculate (a) \(P(X \geq 2)\) and (b) the expected value \(E(X)\).

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

To find \(P(X \geq 2)\), we sum the probabilities of all values of \(X\) that are greater than or equal to 2. So, \(P(X \geq 2) = P(X=2) + P(X=3) + P(X=4) = 0.05 + 0.05 + 0.05 = 0.15\).

Step 3: Calculate Expected Value \(E(X)\)

The expected value \(E(X)\) of a discrete random variable is calculated using the formula \(E(X) = \sum x_i p_i\). Compute this by multiplying each outcome \(x_i\) by its probability \(p_i\) and then summing these products:\[E(X) = (0 \times 0.70) + (1 \times 0.15) + (2 \times 0.05) + (3 \times 0.05) + (4 \times 0.05) = 0 + 0.15 + 0.10 + 0.15 + 0.20 = 0.60\].

Key Concepts

Random Variable

A random variable, in this context, is a numerical representation of outcomes from a random process or experiment. In simple terms, it's like having a special container that holds possible values an experiment can log. Each value is associated with a probability, which helps us understand how likely it is for each event to occur. Consider the random variable \( X \) given in the problem: it can take on one of the discrete set of values \( 0, 1, 2, 3, 4 \). Each of these values has a probability associated with it:

• \( 0 \rightarrow 0.70 \)
• \( 1 \rightarrow 0.15 \)
• \( 2, 3, 4 \rightarrow 0.05 \) each

This distribution tells us how the probabilities spread across the possible outcomes of \( X \). When we name something a discrete random variable, it means it deals with distinct, separate values rather than a continuous range. We use these random variables to model real-life situations, like throwing a dice, and predict outcomes mathematically.

Expected Value

The expected value, often denoted as \( E(X) \), is a fundamental concept in probability. It provides a way to predict the average outcome of the random variable over a large number of trials. Think of it as the long-term average if you were to repeat the random process many times. For our random variable \( X \), the expected value is calculated using the formula:\[ E(X) = \sum x_i p_i \]Where \( x_i \) represents each outcome, and \( p_i \) is the probability of that outcome. To compute this for our given values:

• Multiply each outcome by its probability: \( (0 \times 0.70), (1 \times 0.15), (2 \times 0.05), (3 \times 0.05), (4 \times 0.05) \).
• Add all the products together: \( 0 + 0.15 + 0.10 + 0.15 + 0.20 = 0.60 \).

So, the expected value \( E(X) \) is 0.60, indicating that if you were to observe the outcomes of \( X \) repeatedly, the average result would trend towards 0.6.

Probability Calculation

Probability calculations help us quantify the likelihood of different outcomes for a random variable. In problems involving discrete random variables, like the one given, calculating probabilities often involves summing the weights of individual outcomes.To find \( P(X \geq 2) \), which represents the probability that the random variable \( X \) is greater than or equal to 2, we sum up the probabilities of \( X \) being exactly 2, 3, or 4, because those are the relevant outcomes.

• \( P(X=2) = 0.05 \)
• \( P(X=3) = 0.05 \)
• \( P(X=4) = 0.05 \)

The sum of these probabilities gives us:\[ P(X \geq 2) = 0.05 + 0.05 + 0.05 = 0.15 \]This result indicates that there is a 15% chance that the outcome of \( X \) will be 2 or higher on any single trial of the experiment. These calculations are essential in understanding and predicting outcomes in various probabilistic scenarios in daily life and scientific studies.