Understanding the point estimation of the expected value

Question

I am trying to understand this problem, however I can't get past some of the definitions used when estimating the expected value. What I would need is to confirm or disprove my conclusions - I read school materials and tried to find the answers on internet.

Let's say we measure how much are sea fish toxic and we want to know the expected value of toxicity, when we catch some fish.

From my understanding we should do something like this: If we knew the random variable of toxicity $X$, we could compute the expected value, but we don't know that.

If we want to estimate the expected value, the estimation is: $E(\bar x)= u$, where $\bar x$ is sample mean.

So we catch, for example, $100$ fish and measure the toxicity - now is the part, where I am lost:

In my scripts it says, that this is a random sample, from which we get random vector $(X_1, X_2, X_3, \dots, X_{100})$. How can we get a random vector? Random vector should consist of random variables - does that mean, that each of the fish get's it's own random variable? Or does it mean, that the random vector gives us values of random events and those random events mean : I will catch $100$ of fish from Atlantic is $1$ event and I will catch $100$ fish from Pacific is second event? The second one seems right to me.

Ok let's say we have this random vector
To compute the estimation:
$E(\bar x)=\frac 1n \sum_{i = 1}^{100}x_i$
$N$ = all fish in sea

Now does $x_i$ represent the numerical values of toxicity of each fish caught from our random vector $(X_1, X_2, \dots)$?

So if we would be able to catch all the fish in sea, wouldn't this be just an average?

Maybe the best thing to understand this would be, if someone could estimate the expected value, when for example we caught $10$ fish:
toxicity of each fish: $5,2,7,8,9,1,1,1,2,1 - $ from pool of $100$ fish.

Thanks for replies!

Answer 1

The vector $(x_1,x_2,...,x_{100})$ is our realization of the random vector $(X_1,X_2,...,X_n)$. The fact that $(X_1,X_2,...,X_n)$ is a random vector is simply the fact that every sample of 100 fish is going to give you a different realization of the vector of values $(x_1,x_2,...,x_{100})$. Each $X_i$ is a random variable which represents the value of a random fish. Every time you take a sample of $100$ fish, the toxicity $x_i$ of the $i_{th}$ fish in your sample is random. Thus, the $i_{th}$ fish's toxicity is a random variable $X_i$.

$\bar{X}$ (the sample mean) is a random variable because the sample is itself random. It is our estimator of $\mu$, the average population toxicity. The expression $E(\bar{X})=\mu$ tells us that on average, our sample mean will be equal to the true population toxicity.

Every time we grab 100 fish, we get a different value for the sample mean this is $\bar{x}$. This is our estimate of $\mu$. An estimate is not a random variable, it is just a number. Thus, the expression $E(\bar{x})$ is not very meaningful, the expected value of a number is just the number. $E(\bar{x})=\bar{x}$.

$\bar{x}$ is our estimate of the mean value of toxicity. It is an unbiased estimate since the sample mean as an estimator is on average equal to the true population mean. We have no reason to expect that the sample mean is consistently above or below the true mean.