What Is the Difference Between Expected Value Calculation Methods?

[Peter G.]

Hi,

I have to work with Expected Values and I am extremely confused over the following:

In the part of my book that teaches me about Probability Distribution, in order to calculate the Expected Value I have to:

Lets say we toss a coin twice. We can get 0 Heads, 1 Heads or 2 Heads

I then draw a probability distribution table and the expected value is the sum of the product of the number of heads and their respective probabilities.

When I get to the part that I learn about Binomial Distributions, in order to get the expected value all I have to do is multiply n by p whereas n is the number of tries and p the probability of success.

What is the difference between the two methods? When should I use each?

Thanks!

 

[danago]

Hi,

I have to work with Expected Values and I am extremely confused over the following:

In the part of my book that teaches me about Probability Distribution, in order to calculate the Expected Value I have to:

Lets say we toss a coin twice. We can get 0 Heads, 1 Heads or 2 Heads

I then draw a probability distribution table and the expected value is the sum of the product of the number of heads and their respective probabilities.

When I get to the part that I learn about Binomial Distributions, in order to get the expected value all I have to do is multiply n by p whereas n is the number of tries and p the probability of success.

What is the difference between the two methods? When should I use each?

Thanks!

The equation for the EV of a binomial distribution is derived from the exact same procedure the you have described (sum the product of the outcomes with their respective probabilities) i.e. if X~Bin(n,p), then:

$$E(X)=\sum_{x=0}^{n}xP(X=x)$$

So for any n, you could in fact just draw up a table of outcomes and then continue with how you originally solved it, however that will be a lot more work for large n. It is probably a good exercise to try and actually derive the equation E(X)=np from the equation i have posted above, so you can see for yourself that they are identical.

EDIT:
Have a look at this:

http://amath.colorado.edu/courses/4570/2007fall/HandOuts/binexp.pdf

It will show you the derivation. Also keep in mind that this only works for binomial random variables.