Sage-Tips

How do you calculate conditional expectation of X given Y?

This is called the conditional distribution of X, given that Y = y. Definition: Let X and Y be discrete random variables. The conditional probability function of X, given that Y = y, is: P(X = x|Y = y) = P(X = x AND Y = y) P(Y = y) .

What is the conditional distribution of X given Y Y?

For any random variables X and Y, the conditional distribution of Y given X = x specifies how Y varies when X = x. We have already seen instances of conditional distributions when X and Y are independent. In that case, Y varies just as it usually does, regardless of the values of X.

How do you calculate conditional expectation?

The conditional expectation (also called the conditional mean or conditional expected value) is simply the mean, calculated after a set of prior conditions has happened….Formula and Worked Example

1. 0.03 / 0.49 = 0.061.
2. 0.15 / 0.49 = 0.306.
3. 0.15 / 0.49 = 0.306.
4. 0.16 / 0.49 = 0.327.

What is the expected value of Y given X?

As we will see, the expected value of Y given X is the function of X that best approximates Y in the mean square sense. Note that X is a general random variable, not necessarily real-valued. In this section, we will assume that all real-valued random variables occurring in expected values have finite second moment.

How do you find conditional distribution?

First, to find the conditional distribution of X given a value of Y, we can think of fixing a row in Table 1 and dividing the values of the joint pmf in that row by the marginal pmf of Y for the corresponding value. For example, to find pX|Y(x|1), we divide each entry in the Y=1 row by pY(1)=1/2.

What is the unconditional distribution of Y?

Remarks: •In the example P(H = h | N =n) ~ binomial(n,½). The unconditional distribution of Y is the average of the conditional distributions, weighted by P(X=x). Remark: Once we have the conditional distribution of Y given X=x, using Bayes’ rule we may obtain the conditional distribution of X given Y=y.

What is conditional expectation in statistics?

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of “conditions” is known to occur.

What does conditional mean in statistics?

Conditional probability refers to the chances that some outcome occurs given that another event has also occurred. It is often stated as the probability of B given A and is written as P(B|A), where the probability of B depends on that of A happening.

What is conditional mean in regression?

If you look at any textbook on linear regression, you will find that it says the following: “Linear regression estimates the conditional mean of the response variable.” This means that, for a given value of the predictor variable X , linear regression will give you the mean value of the response variable Y .

How do you calculate conditional CDF?

The conditional CDF of X given A, denoted by FX|A(x) or FX|a≤X≤b(x), is FX|A(x)=P(X≤x|A)=P(X≤x|a≤X≤b)=P(X≤x,a≤X≤b)P(A). Now if x

How do you find conditional CDF?

The conditional CDF of X given A, denoted by FX|A(x) or FX|a≤X≤b(x), is FX|A(x)=P(X≤x|A)=P(X≤x|a≤X≤b)=P(X≤x,a≤X≤b)P(A).

How do you find the conditional expectation of X given y?

If X and Y are discrete random variables , the conditional expectation of X given Y is is the joint probability mass function of X and Y. The sum is taken over all possible outcomes of X . Note that conditioning on a discrete random variable is the same as conditioning on the corresponding event: . Y = y . {\\displaystyle Y=y.}

What is the difference between conditional expectation and random variable?

More formally, in the case when the random variable is defined over a discrete probability space, the “conditions” are a partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted

When is an event with a conditional expectation?

Conditional expectation with respect to an event. In modern [clarification needed] probability theory, when is an event with strictly positive probability, it is possible to give a similar formula. This is notably the case for a discrete random variable and for in the range of if the event is .

How does the σ-algebra affect the conditional expectation?

The σ-algebra controls the “granularity” of the conditioning. A conditional expectation over a finer (larger) σ-algebra retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σ-algebra averages over more events.