# Expected Value of a Discrete Random Variable

## Understanding the Expected Value of a Discrete Random Variable

In the realm of probability and statistics, one encounters a wealth of fascinating concepts and tools that offer profound insights into the uncertain nature of real-world events. Among these, the “Expected Value of a Discrete Random Variable” stands as a fundamental pillar. This concept, often referred to simply as the “Expected Value,” holds the power to illuminate the expected outcomes of uncertain events, making it a cornerstone in decision-making, finance, and various scientific disciplines.

## What is the Expected Value of a Discrete Random Variable?

Expected Value (EV), when applied to a Discrete Random Variable (DRV), provides a numerical representation of the long-term average or expected outcome of a probabilistic event. It offers a way to quantify uncertainty, helping us make informed choices in the face of randomness. Understanding its intricacies is crucial for anyone delving into statistics, probability theory, data science, or fields where uncertainty plays a significant role.

## Why is Expected Value Important?

The concept of Expected Value transcends the confines of academic study; it is a practical tool with real-world implications:

1. Decision-Making: Expected Value serves as a compass in decision-making scenarios. Whether it’s choosing between investment opportunities, insurance policies, or game strategies, knowing the expected outcome helps us make rational choices.
2. Risk Assessment: Businesses and financial institutions rely on Expected Value to assess and manage risks. By quantifying the expected outcomes of various scenarios, organizations can devise strategies to mitigate potential losses.
3. Statistical Analysis: In data analysis, Expected Value plays a pivotal role in summarizing data and making predictions. It forms the basis for various statistical measures, including mean, variance, and covariance.
4. Game Theory: Expected Value is a central concept in game theory, where it aids in determining optimal strategies in competitive situations.

## Calculating the Expected Value

The calculation of Expected Value depends on the nature of the discrete random variable. In this comprehensive guide, we will explore various scenarios, including discrete probability distributions like the Binomial and Poisson distributions, to demonstrate how to calculate the Expected Value effectively.

## Navigating this Guide

To grasp the nuances of the Expected Value of a Discrete Random Variable fully, we’ve structured this guide into distinct sections:

### 1. Understanding Discrete Random Variables (DRV)

• Definition and Characteristics of Discrete Random Variables
• Examples of DRV in Real-World Scenarios

### 2. The Concept of Expected Value

• Definition and Interpretation of Expected Value
• Properties and Significance of Expected Value

### 3. Calculating Expected Value

• Step-by-Step Guide for Finding Expected Value
• Application of Expected Value in Various Probability Distributions

### 4. Practical Applications of Expected Value

• Decision-Making Scenarios
• Financial and Business Applications
• Statistical Analysis and Predictive Modeling

As we delve deeper into these sections, you will gain a comprehensive understanding of the Expected Value of a Discrete Random Variable, its significance, and its versatile applications in diverse fields. By the end of this guide, you’ll be equipped with the knowledge and tools to harness the power of Expected Value in your own analytical endeavours. Let’s embark on this illuminating journey into the world of uncertainty and probability.

For a discrete random variable, the expected value is calculated by summing the product of the value of the random variable and its associated probability, taken over all of the values of the random variable

\large \begin{align} E(aX) &= aE(X) \\ E(X+b) &= E(X) + b \\ E(X^2) &\ne \left[E(X)\right]^2 \end{align}

## Question 1

$\begin{array}{|c|c|c|c|} \hline x & 1 & 2 & 3 \\ \hline \Pr(X=x) & 0.2 & 0.5 & 0.3 \\ \hline \end{array}$

(a)     Find $E(X)$.

\begin{align} &= 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 \\ &= 2.1 \end{align}

(b)     Find $3E(X)$.

\begin{align} &= 3 \times 1 \times 0.2 + 3 \times 2 \times 0.5 + 3 \times 3 \times 0.3 \\ &= 3(1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3) \\ &= 3E(X) \\ &= 3 \times 2.1 \\ &= 6.3 \end{align}

(c)     Find $E(X+2)$.

\begin{align} &= (1+2) \times 0.2 + (2+2) \times 0.5 + (3+2) \times 0.3 \\ &= (1 \times 0.2 + 2 \times 0.2) + (2 \times 0.5 + 2 \times 0.5) + (3 \times 0.3 + 2 \times 0.3) \\ &= (1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3) + 2 \times (0.2+0.5+0.3) \\ &= E(X) + 2 \times 1 \\ &= 2.1 +2 \\ &= 4.1 \end{align}

(d)     Find $E(X^2)$.

\begin{align} &= 1^2 \times 0.2 + 2^2 \times 0.5 + 3^2 \times 0.3 \\ &= 4.9 \end{align}

## Question 2

Given that $E(X)=4$;

(a)     Find $E(3X)$.

\begin{align} &= 3E(X) \\ &= 3 \times 4 \\ &= 12 \end{align}

(b)     Find $E(X+2)$.

\begin{align} &= E(X) +2 \\ &= 4 + 2 \\ &= 6 \end{align}

(c)     Find $E(2X-5)$.

\begin{align} &= 2E(X)-5 \\ &= 2 \times 4-5 \\ &= 3 \end{align}

How to Find Variance using Expected Value of Discrete Random Variables

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