Help with total expectation formula

[yevi]

I need some help with "law of total expectation".

Sorry for my English, I don't know the right English expressions.

The Problem is:
People come (show in) with with Poisson rate of 10 people per hour.
There is a 0.2 chance that a person will give money to a beggar sitting in the corner.
The time that beggar is sitting is U[3,8] (continuous) hours.

Need to find the expectation E of people that will give money.

I mark:
N(t)~P(10*0.2*t) as the number of people giving money per hour.
X~U(3,8) time that beggar is sitting.

The given solution is following : E(N(X))=E(E(N(X)|X))=E(2X)=2*(3+8)/2

What I don't understand is their usage law of total expectation, according to the formula it should be like this:

E(Y)=E[E[Y|X]]=$$\int$$E[Y|X=x]*$$f_{x}$$(x)dx

And this is not what they have...

 

[yevi]

Well, anyone?

 

[Architeuthis Dux]

Hello,

The law of total expectation states that the expected value of a random variable can be calculated by taking the sum of the expected values of the conditional random variables, each multiplied by the probability of their respective events occurring. In this case, the random variable Y represents the number of people giving money, and the random variable X represents the time that the beggar is sitting.

The formula that you have mentioned is correct, but it is applied differently in this problem. The given solution uses the law of total expectation by first finding the conditional expected value of N(X) given X, denoted as E(N(X)|X). This is then multiplied by the probability density function of X, f(x), and integrated over the range of X to get the overall expected value of N(X).

In other words, E(N(X))=E[E(N(X)|X)]=∫E(N(X)|X=x)*f(x)dx.

In this case, the conditional expected value E(N(X)|X) is equal to 2X, as each person has a 0.2 probability of giving money and X represents the time in hours that the beggar is sitting. Therefore, the overall expected value of N(X) is 2X multiplied by the probability density function of X, which is 1/5 for a uniform distribution from 3 to 8. This gives us E(N(X))=2X*(1/5)=2X/5.

To find the expected value of X, we can use the formula E(X)=∫x*f(x)dx. In this case, the probability density function of X is 1/5 for a uniform distribution from 3 to 8. Therefore, E(X)=∫x*(1/5)dx=5.5.

Substituting this value into our previous equation, we get E(N(X))=2*5.5/5=11/5=2.2.

I hope this helps clarify the usage of the law of total expectation in this problem. Let me know if you have any further questions.