Problem 188 - An Introduction to Mathematical Statistics and Its Applications

Question

If a random variable \(V\) is independent of two independent random variables \(X\) and \(Y\), prove that \(V\) is independent of \(X+Y\).

Step-by-Step Solution

Answer

The exercise is proven by applying the benefits of the independence of \(X\), \(Y\) and \(S\) on each other in addition to the independence of \(V\) from \(X\) and \(Y\). After careful application of these facts, it was successfully proven that \(V\) is independent of \(X + Y\).

Step 1: Definition of Independence

Let's recall the definition of two independent random variables \(X\) and \(Y\): they are independent if and only if, for each pair of x and y, \(P(X = x, Y = y) = P(X = x)P(Y = y)\). Likewise, if \(V\) is independent of \(X\) and \(Y\), we have \(P(V = v, X = x) = P(V = v)P(X = x)\) and \(P(V = v, Y = y) = P(V = v)P(Y = y)\).

Step 2: Expressing \(X+Y\)

We are intended to prove \(V\) is independent of \(X + Y\). Note that 'independence' implies that knowing the outcome of one random variable does not influence the probability of the outcome of the other variable. However, we don't know if that statement is true for the sum of \(X + Y\). Let's define \(S = X + Y\). Since \(X\) and \(Y\) are independent, then \(P(S = s) = \sum{P(X=x, Y=s-x)} = \sum{P(X=x)P(Y=s-x)}\).

Step 3: Proving The Independence Of \(V\) And \(X+Y\)

We want to show that \(P(V = v, S = s) = P(V = v) P(S = s)\). So, let's compute \(P(V = v, S = s) = P(V = v, X = x, Y = s-x) = P(X = x)P(Y = s - x)P(V = v) = P(V = v)P(S = s)\). And it proves that \(V\) is independent of \(X + Y\) (or \(S\)).

Key Concepts

Random Variable

A random variable is an abstract concept used in probability and statistics. It is a function that assigns a numerical value to each possible outcome of a random phenomenon.
Random variables can be discrete or continuous. A discrete random variable can take on a countable number of values, while a continuous random variable can take on an infinite number of values within a given range.

• Discrete random variables often arise in scenarios where outcomes are counted, like the toss of a dice.
• Continuous random variables are used when outcomes are measured, like the time taken to run a race.

Understanding random variables is crucial for calculations involving probability distributions, expected values, and variances. They help transform real-world phenomena into a mathematical framework that can be analyzed and studied.
By assigning numerical values to outcomes, random variables allow us to use mathematical tools to predict and understand patterns and behaviors in uncertain events.

Probability

Probability is the mathematical study of randomness and uncertainty. It provides a measure of how likely an event is to occur, expressed as a number between 0 and 1. A probability of 0 indicates impossibility, while 1 indicates certainty.
Calculating probability involves considering all possible outcomes of an event and determining how many of these outcomes satisfy a certain condition.

• For discrete scenarios, probabilities are often calculated by counting outcomes and dividing by the total number of possible outcomes.
• For continuous scenarios, probabilities involve integrating over a range of values.

Probability is also defined through probability distributions, such as the normal distribution or binomial distribution, which describe how probabilities are distributed across different outcomes.
Understanding probability is essential for fields like statistics, economics, and science, as it underpins the ability to make informed predictions and decisions based on uncertain information.

Independence

The concept of independence in probability is crucial for understanding how events or random variables relate to each other. Two random variables are said to be independent if the occurrence or outcome of one does not affect or influence the occurrence of the other.
In mathematical terms, for two random variables, say \( X \) and \( Y \), they are independent if:\[P(X = x, Y = y) = P(X = x) P(Y = y)\]This equation states that the probability of both \( X \) and \( Y \) occurring simultaneously is simply the product of their individual probabilities.

• Independence is a vital property in statistics because it simplifies the analysis and calculation of probabilities involving multiple variables or events.
• When random variables \( X \) and \( Y \) are independent, knowing the outcome of \( X \) gives no information about \( Y \), and vice versa.

In the specific context of the problem, demonstrating that a random variable \( V \) is independent of \( X+Y \) relies on this fundamental property. Given \( V \) is independent of two independent variables \( X \) and \( Y \), it maintains independence from their sum, confirming that the combined effect of \( X \) and \( Y \) does not influence \( V \). This illustrates the broader application of independence among variables in probability.