Recall that two events \(A\) and \(B\) are called independent if \(p(A B)=p(A) \cdot p(B) .\) Similarly, two random variables \(x\) and \(y\) with probability functions \(f(x)\) and \(g(y)\) are called independent if the probability of \(x=x_{i}\) and \(y=y_{j}\) is \(f\left(x_{i}\right) \cdot g\left(y_{j}\right)\) for every pair of values of \(x\) and \(y\), that is, if the joint probability function for \(x, y\) is \(f(x) g(y)\). Show that if \(x\) anct \(y\) are independent, then the expectation or average of \(x y\) is \(E(x y)=E(x) \cdot E(y)=\mu_{x} \mu_{y}\).

Short Answer

Expert verified
If x and y are independent, then E(xy) = E(x) * E(y).

Step by step solution

01

Understand the Given Information

We are given two random variables, x and y, that are independent. The goal is to show that the expectation of their product, E(xy), is equal to the product of their individual expectations, E(x) and E(y).
02

Write Down the Definition of Independence

For two random variables x and y to be independent, their joint probability function must be the product of their individual probability functions: \[ p(x_i, y_j) = f(x_i) \times g(y_j) \]
03

Formulate the Expectation of xy

The expectation of the product of two random variables is defined as: \[ E(xy) = \sum_{i,j} x_i y_j p(x_i, y_j) \]
04

Substitute the Joint Probability Function

Since x and y are independent, we can write: \[ E(xy) = \sum_{i,j} x_i y_j [f(x_i) \times g(y_j)] \]
05

Separate the Sums

Since the sums are over independent terms, we can separate them: \[ E(xy) = \sum_{i} x_i f(x_i) \times \sum_{j} y_j g(y_j) \]
06

Recognize Expectations in the Result

Notice that: \[ \sum_{i} x_i f(x_i) = E(x) \text{ and } \sum_{j} y_j g(y_j) = E(y) \]So we get: \[ E(xy) = E(x) \times E(y) \]
07

State the Final Result

Thus, we have shown that if x and y are independent random variables, the expectation of their product is: \[ E(xy) = E(x) \times E(y) = \mu_{x} \mu_{y} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

random variables independence
Let's dive into the concept of independence for random variables.
Two random variables, say \(x\) and \(y\), are considered independent if knowing the value of one does not give any information about the other.
In mathematical terms, this means their joint probability function can be factored into the product of their individual probability functions:
  • The joint probability function \(p(x_i, y_j)\) represents the probability that \(x = x_i\) and \(y = y_j\) occur simultaneously.
  • If \(x\) and \(y\) are independent, then \(p(x_i, y_j)\) can be written as the product of the marginal probabilities \(f(x_i)\) and \(g(y_j)\), where \(f(x_i)\) and \(g(y_j)\) are the probabilities of \(x = x_i\) and \(y = y_j\) occurring independently.
To put it simply, for independent events: \[ p(x_i, y_j) = f(x_i) \times g(y_j) \]
joint probability function
Now, let’s explore the joint probability function a bit more.
A joint probability function \(p(x, y)\) describes the probability of random variables \(x\) and \(y\) taking on specific pairs of values \((x_i, y_j)\). For independent random variables, as discussed, this joint function decomposes:
  • The joint probability distribution simplifies to the product of individual (marginal) distributions.
  • Thus, if \(x\) and \(y\) are independent, we have: \( p(x_i, y_j) = f(x_i) \times g(y_j) \).

Intuitively, this means we are treating the occurrence of \(x = x_i\) and \(y = y_j\) as separate events that don’t influence each other.
Mathematically, this laying apart the joint probability allows us to work with each variable individually without worrying about their interdependence in data-driven contexts.
expectation of product
Lastly, let’s tackle the expectation of the product of two independent random variables.
When we want to find the expected value (or mean) of the product \(xy\), we use the definition of expectation:
\[ E(xy) = \sum_{i,j} x_i y_j p(x_i, y_j) \]
Given that \(x\) and \(y\) are independent, we can substitute their joint probability function \(p(x_i, y_j)\) with \( f(x_i) \times g(y_j) \).
This gives us:
\[ E(xy) = \sum_{i,j} x_i y_j [f(x_i) \times g(y_j)] \]
We can now separate the sums as follows:
\[ E(xy) = \sum_{i} x_i f(x_i) \times \sum_{j} y_j g(y_j) \]
Recognize that \( \sum_{i} x_i f(x_i) \) is the definition of \(E(x)\) and \( \sum_{j} y_j g(y_j) \) is the definition of \(E(y)\). Hence, we arrive at our result:
\[ E(xy) = E(x) \times E(y) \]
So, because of their independence, the expectation of the product of \(x\) and \(y\) equals the product of their expectations. This is a crucial insight, making calculations involving expected values more manageable in probabilistic analysis.

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