# #StackBounty: #estimation #sampling #sample #consistency Which is the relation between population/probability space/sampling?

### Bounty: 50

I am trying to understand the relation between population/probability space/sampling. My arguments are divided in 3 sub-questions which trace my attempt to link in a logical way the three concepts. I am using an Economic example, but I regard my question as generic.

Consider a target population of individuals; for each individual in this population we believe that \$\$income=beta*education+u\$\$

where \$income, education, u\$ are real numbers. \$u\$ collects all the variables affecting income in addition to education.

We are interested in learning about \$beta\$.

Sub-question 1: Imagine to extract at random an individual \$m\$ from the population and observe her income, education, and additional features (denoted by the random variables \$Y_m, X_m, U_m\$). Intuitively, I understand why we can say that the income, education, additional features of the extracted individual are random variables: we don’t know a priori which individual will be picked from the urn representing the whole population and, hence, we attach a probability to each potential outcome. More formally, to define a random variable we need a probability space \$(Omega, mathcal{F}, Pr)\$. Is \$Omega\$ set equal to the population?

Sub-question 2: If the answer to question 1 is YES, then, if we knew the entire population (i.e., if we knew \$(Omega, mathcal{F}, Pr)\$) and suppose \$E(X_m^2)neq 0\$ and \$E(X_mU_m)=0\$, we could easily compute the exact value of \$beta\$ as
\$\$
beta=E(Y_mX_m)/E(X_m^2)
\$\$
Correct?

Sub-question 3: The problem is that we don’t know the entire population (i.e., we don’t know \$(Omega, mathcal{F}, Pr)\$) and so we try to approximate in some good way \$E(Y_mX_m)\$ and \$E(X_m^2)\$ by appropriately taking a subset of the whole population (sampling). For example, a way to appropriately take a subset of the whole population is the following: for \$m=1,…,M\$:

• We draw at random an individual from the urn containing the entire population, we label him/her with the index \$m\$, and we register his/her income and education level (denoted by the random variables \$Y_m, X_m)\$. The additional features affecting income remain unobserved (denoted by the random variables \$U_m\$).

• We put back in the urn individual \$m\$.

The sampling scheme just described implies that
\$\$
(i) hspace{1cm}{Y_m, X_m, U_m}_{m=1}^M text{ are i.i.d. across \$m\$}
\$\$

We then define
\$\$
hat{beta}=frac{frac{1}{M}sum_{m=1}^MY_mX_m}{frac{1}{M}sum_{m=1}^M X_m^2}
\$\$
By \$(i)\$, \$frac{1}{M}sum_{m=1}^MY_mX_mrightarrow_p E(Y_mX_m)\$ and \$frac{1}{M}sum_{m=1}^M X_m^2rightarrow_p E(X^2_m)\$.

Hence,
\$\$
hat{beta}rightarrow_p beta
\$\$
Correct?

Sub-question 4: Suppose also that \$U_m\$ is a continuous random variable. Does this imply stating that the population is very large or a continuum or infinite?

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