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Magazine Chapter 3: Problem 29
If \(Y\) is a discrete random variable that assigns positive probabilities to only the positive integers, show that $$E(Y)=\sum_{i=1}^{\infty} P(Y \geq k)$$.
Short Answer
Expert verified
The expected value \( E(Y) = \sum_{k=1}^{\infty} P(Y \geq k) \) is derived by expressing it using cumulative probabilities and simplifying.
Step by step solution
01
Understand the Expectation of a Discrete Random Variable
For a discrete random variable \( Y \) that takes on integer values, the expected value, or mean, is defined as \( E(Y) = \sum_{k=1}^{\infty} k \cdot P(Y=k) \). This is a weighted average of the possible values of \( Y \), where each value is weighted by its probability.
02
Rewrite Expectation Formula with Cumulative Probability
Consider the probability that \( Y \) is at least \( k \), which is \( P(Y \geq k) \). Notice that \( k \cdot P(Y=k) = \sum_{i=1}^{k} P(Y \geq i) - (k-1) \cdot P(Y = k) \). By rearranging terms and using the formula \( k \cdot P(Y=k) = \sum_{i=1}^{k} (P(Y \geq i) - P(Y \geq i+1)) \), we can express the expected value in terms of cumulative probabilities.
03
Express Total Expectation as a Sum of Cumulative Probabilities
Substitute \( k \cdot P(Y=k) = \sum_{i=1}^{k} (P(Y \geq i) - P(Y \geq i+1)) \) into the expectation formula: \( E(Y) = \sum_{k=1}^{\infty} \left[\sum_{i=1}^{k} (P(Y \geq i) - P(Y \geq i+1)) \right] \). Upon rearranging terms, you end up summing \( P(Y \geq i) \) infinitely over all positive integers \( i \).
04
Simplify to Final Conclusion
In the final arrangement, sums form a telescoping series, which simplifies to \( E(Y) = \sum_{k=1}^{\infty} P(Y \geq k) \). The inner terms cancel out sequentially, leaving only a sum over cumulative probabilities for each positive integer \( k \).
05
Conclusion
We have shown that the expected value of a discrete random variable \( Y \) with positive integer values is given by \( E(Y) = \sum_{k=1}^{\infty} P(Y \geq k) \). This approach exploits the relation between distribution tails and expected values.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Expected Value
Expected value is a fundamental concept in probability and statistics, particularly for understanding the behavior of discrete random variables. It is a measure of the central tendency, analogous to the mean in descriptive statistics. For a discrete random variable, the expected value represents the long-term average outcome if an experiment is repeated many times. It is calculated by:\[E(Y) = \sum_{k=1}^{\infty} k \cdot P(Y=k)\]Here, each possible value of the random variable, denoted by \( k \), is multiplied by its probability, \( P(Y=k) \), and the results are summed over all possible values.
- **Discrete random variable:** A variable that can take on a finite or countably infinite set of values, like rolling a die.
- **Central tendency:** The expected value serves as a typical value describing a distribution.
- **Weighted average:** The values of the variable are multiplied by their probabilities, thus each is weighted by its likelihood.
Cumulative Probability
Cumulative probability is a way to consider all outcomes at or below a particular value. For discrete random variables, it often helps to simplify complex probability problems. When dealing with the expectation of a random variable, we can use cumulative probabilities instead of point probabilities to find interesting relations.Consider the probability that the random variable \( Y \) is greater than or equal to some positive integer \( k \), written as \( P(Y \geq k) \). This tells us the likelihood of \( Y \) taking on values starting from \( k \) upward.
- **Cumulative probability:** Summing probabilities from a baseline to a certain point.
- **Rewriting equations:** Using cumulative probability in expectation equations can lead to simpler forms and solutions.
Probability
Probability is the mathematical foundation of randomness and chance. It describes the likelihood of events occurring and is an essential part of understanding random variables. With respect to discrete random variables, each outcome has an associated probability, \( P(Y = k) \), for value \( k \).The probabilities of all possible outcomes must sum to one, which maintains the integrity and balance of the probabilistic model.
- **Discrete probabilities:** Opportunity for events that can be distinctly counted.
- **Probability distribution:** A list showing all possible values and their probabilities.
- **Key properties:** Non-negative probabilities adding to one.
Positive Integers
Positive integers are essential when discussing certain types of random variables, particularly those who only take on values like 1, 2, 3, and so forth. These values, without any zero or negative numbers, are vital when addressing discrete random variables in many real-world and theoretical scenarios.In the context of the problem, the variable \( Y \) assigns probabilities only to these positive integers and not to zero or any negative numbers. This constraint helps to frame both the probability distribution and the expected value in a simpler way.
- **Positive integers:** Numbers greater than zero which can be counted.
- **Impact on equations:** Ensure calculations focus on meaningful and practical outcomes.
- **Discrete variables:** Often supported by positive integer values.
