Multivariate probability distributions?

In summary, the conversation is about finding the mean and variance of the number of customers who open a new ABC bank account in a day. It is known that the number of customers entering the bank follows a Poisson distribution with an unknown mean lambda. After discussing different approaches, it is determined that the number of customers who open a new account can be represented by a Poisson distribution with a mean of 0.01 times lambda. The final conclusion is that the mean is 0.01 times lambda and the variance is 0.0001 times lambda. However, the value of lambda is still unknown.
  • #1
kingwinner
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Homework Statement


Let Y be the number of customers entering a ABC bank in a day. It is known that Y has a Poisson distribution with some unknown mean lambda. Suppose that 1% of the customers entering the branch in a day open a new ABC bank account. Find the mean and variance of the number of customers who open a new ABC bank account in a day.


Homework Equations


The Attempt at a Solution



Just based on past experience, I THINK this is related to multivariate distributions and PROBABLY would use the fact E[E(Y|X)]=E(Y). But I am not sure how to define the random variables properly...

Can someone explain?

Thanks for any help!
 
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  • #2
No, it is NOT a multivariate distribution- there is only the single variable, the number of customers. You are asked for the mean and variance of 0.01Y where Y is Poisson distributed.

If [itex]\sum yP(y)= \lambda[/itex], what is [itex]\sum 0.01 yP(y)[/itex]?
 
  • #3
OK, so this is actually a univariate problem...

Let Z=number of customers who open a new ABC bank account in a day
Z~Poisson(0.01*lambda)
Then our job is to find E(Z) and Var(Z)?

Or is our job to find E(0.01Y) and Var(0.01Y) where Y~Poisson(lambda)?

Thank you!
 
  • #4
So Z = 0.01 Y, we need to find E(Z) and Var(Z)
E(Z)=E(0.01Y)=0.01E(Y)=0.01*lambda
Var(Z)=Var(0.01Y)=(0.01^2) Var(Y)=0.0001*lambda

Am I right? Is to any way to find lambda?
 

What is a multivariate probability distribution?

A multivariate probability distribution is a statistical model that describes the probabilities of multiple variables occurring together. It takes into account the relationships between the variables and their individual probabilities.

What are the applications of multivariate probability distributions?

Multivariate probability distributions have a wide range of applications in various fields, such as finance, economics, biology, and engineering. They are useful in analyzing complex data sets and making predictions based on multiple variables.

How is a multivariate probability distribution different from a univariate distribution?

A univariate distribution describes the probabilities of a single variable, while a multivariate distribution describes the probabilities of multiple variables occurring together. In a multivariate distribution, the variables are often dependent on each other, whereas in a univariate distribution, the variable is independent.

What are some common types of multivariate probability distributions?

Some common types of multivariate probability distributions include the multivariate normal distribution, the multivariate binomial distribution, and the multivariate Poisson distribution. Each of these distributions has its own set of properties and is used in different situations.

What are the challenges in working with multivariate probability distributions?

One of the main challenges in working with multivariate probability distributions is the complexity of the data. It can be difficult to analyze and interpret the relationships between multiple variables. Additionally, the calculations involved in working with multivariate distributions can be more complex and time-consuming compared to univariate distributions.

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