Problem 7 - 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 \((b) E(X)\). $$ p_{i}=(5-i) / 10, x_{i}=i, i=1,2,3,4 $$

Step-by-Step Solution

Answer

(a) 0.6, (b) 2.0

Step 1: Listing the Probabilities

First, let's determine the probabilities for each value of the random variable \(X\). Based on the given distribution, we have:- For \(i = 1\), \(p_{1} = \frac{5-1}{10} = 0.4\),- For \(i = 2\), \(p_{2} = \frac{5-2}{10} = 0.3\),- For \(i = 3\), \(p_{3} = \frac{5-3}{10} = 0.2\),- For \(i = 4\), \(p_{4} = \frac{5-4}{10} = 0.1\).

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

To find \(P(X \geq 2)\), we must sum the probabilities where \(X\) is 2 or greater. Therefore:\[ P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.3 + 0.2 + 0.1 = 0.6 \]

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

The expected value \(E(X)\) is calculated by multiplying each value of \(X\) by its probability and summing these products:\[ E(X) = x_{1}p_{1} + x_{2}p_{2} + x_{3}p_{3} + x_{4}p_{4} \]Substitute the values:\[ E(X) = 1 \times 0.4 + 2 \times 0.3 + 3 \times 0.2 + 4 \times 0.1 = 0.4 + 0.6 + 0.6 + 0.4 = 2.0 \]

Key Concepts

Expected Value

The expected value, often represented as \(E(X)\), is a fundamental concept in probability and statistics. It essentially tells us what the average outcome of a random variable should be if we were to repeat an experiment many times under the same conditions. For a discrete random variable, the expected value is calculated by multiplying each possible outcome by its probability, and then adding all those products together. In mathematical terms:\[ E(X) = \sum (x_i \cdot p_i) \]where \(x_i\) is each possible value that the random variable can take, and \(p_i\) is the probability that \(X\) equals \(x_i\). In the context of our exercise, this calculation shows us that if the scenario was repeated over and over, on average, you would expect the outcome to be 2.0. This average, or expected value, suggests the central tendency around which the data points, or outcomes, tend to cluster.

Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In simple terms, it's a way to measure the outcomes with numbers. Random variables can be:

• Discrete - where the variable can take on only specific, separate values (like counting whole numbers)
• Continuous - where the variable can take on any value within a range (like measuring time or distance)

In our exercise, the random variable \(X\) is discrete, meaning it can take on specific values (1, 2, 3, or 4) based on the given probability distribution. Each possible value of \(X\) has its own probability, which we calculated using the given formula \(p_i = \frac{5-i}{10}\).Understanding the concept of random variables is vital because it forms the basis for describing distributions, probability, and various statistical analyses.

Probability Calculation

Probability calculation is at the heart of statistics, allowing us to quantify the likelihood of different outcomes. To calculate the probability of a specific event, we use the basic probability formula:\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]In discrete distributions, each outcome for a random variable has a specific probability. Using these, we can calculate probabilities for more complex scenarios like combined events.In part (a) of the exercise, we calculated \(P(X \geq 2)\) by adding the probabilities of the events where \(X\) equals 2, 3, and 4:\[ P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.3 + 0.2 + 0.1 = 0.6 \]This tells us there is a 60% chance the random variable will take a value of 2 or greater. Understanding this step helps in grasping how probabilities can be combined to solve more complex events.