Question

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find \((a)\) \(P(X \geq 2)\) and \((b) E(X)\). $$ \begin{array}{l|llll} x_{i} & 1 & 2 & 3 & 4 \\ \hline p_{i} & 0.4 & 0.2 & 0.2 & 0.2 \end{array} $$

Short answer

(a) 0.6; (b) 2.2

Step-by-step solution

Identify Relevant Probabilities for P(X ≥ 2)

To find \( P(X \geq 2) \), sum the probabilities for the values of \( X \) that satisfy the condition \( X \geq 2 \). The relevant values are 2, 3, and 4.

Calculate P(X ≥ 2)

Using the probabilities given: \( P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.2 + 0.2 + 0.2 = 0.6 \).

Understand Expectation Formula

The expected value \( E(X) \) is calculated using the formula: \( E(X) = \sum (x_i \cdot p_i) \), where \( x_i \) is each value of X and \( p_i \) is the corresponding probability.

Calculate E(X)

Calculate \( E(X) \) step-by-step: - For \( x_1 = 1 \), \( 1 \cdot 0.4 = 0.4 \) - For \( x_2 = 2 \), \( 2 \cdot 0.2 = 0.4 \) - For \( x_3 = 3 \), \( 3 \cdot 0.2 = 0.6 \) - For \( x_4 = 4 \), \( 4 \cdot 0.2 = 0.8 \) Then sum these products: \( 0.4 + 0.4 + 0.6 + 0.8 = 2.2 \).

Verification

Ensure that the probability sum is 1: \( 0.4 + 0.2 + 0.2 + 0.2 = 1 \). This confirms that the distribution is valid.

Key concepts

Discrete Random Variable

A discrete random variable is a type of variable that takes on specific, isolated values. Unlike continuous variables, which can take any value within a range, discrete random variables have distinct, separate values. In many ways, it's like counting whole numbers. For example, the number of books in a backpack can be 0, 1, 2, etc., but it cannot be 1.5. This concept is important as it helps us understand situations where outcomes are not fluid but fixed. In probability distributions, a discrete random variable has associated probabilities for each value it can take. That means each possible outcome has its probability, and these must be pre-defined for calculating things like expected value and probability of certain events.

Expected Value

The expected value is a valuable concept in probability theory. It's like the average result you would expect if you could repeat a random process an infinite number of times. Although each trial might yield different results, the expected value gives a long-term average.To calculate it for a discrete random variable, use this formula:\[E(X) = \sum (x_i \cdot p_i)\]Here, \(x_i\) represents each possible value the random variable can take, and \(p_i\) is the probability of each \(x_i\). This weighted average helps us determine the central tendency of a random variable.In the exercise example, the expected value was calculated as 2.2, showing that if you repeatedly sample from this distribution, the average of your results would tend toward 2.2.

Sum of Probabilities

The sum of probabilities is an essential feature of any probability distribution. For a distribution to be valid, the probabilities of all possible outcomes must add up to 1. This represents the fact that one of the outcomes must occur.Whenever you are given a set of probabilities, it's a good practice to add them together to ensure that they sum to 1. This ensures you've included all potential outcomes and that the probability distribution is complete.In the given problem, the sum of the probabilities \(0.4 + 0.2 + 0.2 + 0.2 = 1\) verifies that it is a valid probability distribution. Ensuring this sum equals 1 is a fundamental check to confirm the distribution is accurate.

Valid Probability Distribution

A valid probability distribution is important because it ensures that the probabilities assigned to all potential outcomes make logical sense. For a distribution to be valid:- All probabilities must be between 0 and 1.- The sum of all probabilities must be exactly 1.This guarantees that no invalid probabilities are used, such as negative probabilities or probabilities greater than 1.In the exercise, each \(p_i\) was between 0 and 1, and their sum equaled 1, confirming it's a valid probability distribution. Understanding this concept helps in checking whether a given distribution can be legitimately used to calculate probabilities or expected values. Valid distributions form the basis of making accurate and meaningful inferences from data.