# Conditional expectation of joint PMF

• MHB
• nacho-man
In summary, The conversation is about clarifying the difference between E[X1 X2] and E[X1 | X2] and how to compute them. The person asking the question is struggling with understanding the notation and procedure for computing conditional expectations for discrete random variables. They also ask for help in determining if two random variables, X1 and X2, are independent. The expert provides a summary of the notation and procedure for computing E[X1 | X2] and explains how to determine if X1 and X2 are independent. They also mention that the conversation answers all the questions asked.
nacho-man
The question appears to be simple enough, but i have two queries

A) does E[X1 X2] mean the same as E[X1 | X2]

B) If not/so, how exactly do I go about computing this. I've seen a few formulas in my lectures notes for computing conditional expectations for discrete random variables,
however I find it difficult to understand and apply the notation/procedure.

Any help is appreciated!

edit: ok, after some more research, I've found that
E(X1 X2] simply means The expectations of X1 and X2 multiplied by each other.

so, what I want to ask now is this.
is the PMF of X1, given that table:

X1 | -1 | 0 | 1 |
px(X1)| 1/3 | 0 | 1/3 |
And finally, how do i find out if X1 and X2 are independent?

EDIT 2: okay, is this correct

for E[X1 X2]
i do:

(-1)(-1)*(1/6) + ...

That is multiply each (X1,X2) and then multiply that by the probability of its occurrence, and add them all up?

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nacho said:
okay, is this correct

for E[X1 X2]
i do:

(-1)(-1)*(1/6) + ...

That is multiply each (X1,X2) and then multiply that by the probability of its occurrence, and add them all up?

Yes.

If not, then let me add that to find E[X1|X2=1] you find each pair (x1,x2) such that x2=1 and do:
((-1)*(1/6) + ...) / ((1/6) + ...)

nacho said:
The question appears to be simple enough, but i have two queries

A) does E[X1 X2] mean the same as E[X1 | X2]

B) If not/so, how exactly do I go about computing this. I've seen a few formulas in my lectures notes for computing conditional expectations for discrete random variables, however I find it difficult to understand and apply the notation/procedure.

Any help is appreciated!...

In formal notation $\displaystyle E [X_{1}\ X_{2}]$ means the expected value of the r.v. $\displaystyle X = X_{1}\ X_{2}$ and $\displaystyle E [X_{1} | X_{2}]$ means the expected value of the r.v. $\displaystyle X_{1}$ conditioned by the r.v. $\displaystyle X_{2}$, i.e... $\displaystyle E [X_{1}| X_{2}] = \sum x_{1} P \{X_{1}=x_{1} | X_{2} = x_{2}\}$

Kind regards

$\chi$ $\sigma$

## What is the definition of conditional expectation of joint PMF?

The conditional expectation of joint PMF refers to the expected value of a random variable given the value of another random variable. It is a measure of the average value of one variable when the other variable is held constant.

## How is conditional expectation of joint PMF calculated?

The conditional expectation of joint PMF can be calculated by taking the sum of the product of each possible value of the first random variable and its corresponding conditional probability, given a specific value of the second random variable.

## What is the difference between conditional expectation of joint PMF and marginal PMF?

The conditional expectation of joint PMF takes into account the value of another random variable, while marginal PMF only considers the probability of one single variable. Conditional expectation can be seen as a more specific version of marginal PMF.

## What is the significance of conditional expectation of joint PMF in statistics?

Conditional expectation of joint PMF is an important concept in statistics as it allows for the analysis of the relationship between two random variables. It can help in making predictions and understanding patterns in data.

## Can conditional expectation of joint PMF be applied to continuous random variables?

Yes, conditional expectation of joint PMF can be applied to both discrete and continuous random variables. However, the calculations may differ for continuous variables due to the use of integrals instead of summation.

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