# Conditional Epectation of Multinomial

• torquerotates
In summary, To find E(X1,X2|Xn) for a multinomial distribution with non-independent observations, the distribution of (X1, X2| Xn = 6) would be a multinomial distribution for 6 less objects with the cell probabilities for X1, X2,..X[n-1] scaled by a factor so they sum to 1. The notation E(X1,X2) represents the expectation of x1 and x2.

#### torquerotates

So I'm trying to find E(X1,X2|Xn) where X1,X2,...Xn are the numbers of cell observations in a multinomial distribution. How do I even calculate this? I know it is not independent so I cannot split it.

Does it have something to do with the fact that E(Xi)=nPi ?

Intuitively, we would expect the distribution of $(X1, X2| Xn = 6)$ to be a multinomial distribution for 6 less objects with the cell probabilities for $X1, X2,..X[n-1]$ scaled by a factor so they sum to 1. I don't know what your notation $E(X1,X2...$ means. Are you asking about a vector of expectations?

Stephen Tashi said:
Intuitively, we would expect the distribution of $(X1, X2| Xn = 6)$ to be a multinomial distribution for 6 less objects with the cell probabilities for $X1, X2,..X[n-1]$ scaled by a factor so they sum to 1. I don't know what your notation $E(X1,X2...$ means. Are you asking about a vector of expectations?

Interesting. Makes sense.

Well, E(X1,X2) is just expectation of x1 and x2

## 1. What is conditional expectation of multinomial?

Conditional expectation of multinomial is a statistical concept that refers to the expected number of successes in a multinomial experiment given certain conditions or events. It is used to predict the outcomes of a multinomial experiment based on prior knowledge or information.

## 2. How is the conditional expectation of multinomial calculated?

The conditional expectation of multinomial is calculated by taking the product of each possible outcome with its corresponding probability, and then summing these products. This can be represented mathematically as E[X|A] = ∑x P(X=x|A), where X is the random variable, A is the event, and P(X=x|A) is the probability of X being equal to x given event A.

## 3. What is the difference between conditional expectation of multinomial and unconditional expectation of multinomial?

The unconditional expectation of multinomial refers to the expected number of successes in a multinomial experiment without any prior knowledge or information, while the conditional expectation of multinomial takes into account certain conditions or events and predicts the expected number of successes accordingly.

## 4. What is the application of conditional expectation of multinomial in real life?

The conditional expectation of multinomial has various applications in real life, such as in market research, where it can be used to predict the success of a product based on certain demographics or characteristics of the target market. It is also used in sports analytics to predict the outcome of a game given certain conditions, such as player injuries or weather conditions.

## 5. How is the conditional expectation of multinomial used in decision-making?

The conditional expectation of multinomial is often used in decision-making processes to help weigh different options and make informed decisions. By taking into account certain conditions or events, it can provide a more accurate prediction of the expected outcomes, helping individuals or organizations make better decisions.