- #1
mnb96
- 715
- 5
Hello,
I have a discrete random variable z whose expected value [itex]\mu[/itex] is unknown. Its distribution is also unknown.
We extract N samples [itex]z_1,\ldots,z_N[/itex], where each sample is an integer number: [itex]z_i\in \mathbb{Z}[/itex].
Now, I introduce an estimator for the expected value defined as follows:
[tex]\overline{z}=\frac{1}{N}\sum_{i=1}^N z_i[/tex]
How should I compute the expected value [itex]E[\overline{z}][/itex] of the estimator [itex]\overline{z}[/itex]?
***
In my lecture notes I read:
[tex]E[\overline{z}]=E[\frac{1}{N}\sum_{i=1}^N z_i]=\frac{1}{N}\sum_{i=1}^NE[z_i]=\mu[/tex]
This makes no sense to me, especially the term [itex]E[z_i][/itex].
What is supposed to represent the expected value of an observation?!
I have a discrete random variable z whose expected value [itex]\mu[/itex] is unknown. Its distribution is also unknown.
We extract N samples [itex]z_1,\ldots,z_N[/itex], where each sample is an integer number: [itex]z_i\in \mathbb{Z}[/itex].
Now, I introduce an estimator for the expected value defined as follows:
[tex]\overline{z}=\frac{1}{N}\sum_{i=1}^N z_i[/tex]
How should I compute the expected value [itex]E[\overline{z}][/itex] of the estimator [itex]\overline{z}[/itex]?
***
In my lecture notes I read:
[tex]E[\overline{z}]=E[\frac{1}{N}\sum_{i=1}^N z_i]=\frac{1}{N}\sum_{i=1}^NE[z_i]=\mu[/tex]
This makes no sense to me, especially the term [itex]E[z_i][/itex].
What is supposed to represent the expected value of an observation?!