Population Cumulative Distribution Function with Simple Random Sampling

Question

Link to textbook example: http://puu.sh/CEgk2/b5d08453fb.png

Context to Example 5.4.1: http://puu.sh/CEgmt/c6390a8bba.png

Hi, I was reading up on my textbook readings for my intro to stats class, and I got confused over an example that was provided in the textbook. I was wondering if I could get some help with understanding it.

In the first picture, there is an equation that is equivalent to CDF. Please correct me if I'm understanding this wrong, but the first part of that equation says "the probability of the measurement X of the first individual in the population being less than x is equal to...". Then, the second part of the equation says "the absolute value of the number of individuals from the population set such that the measurement of the population is less than or equal to x over 20". Then, the third part is just the CDF. The thing that confuses me is that if the probability is being asked of the first individual of the population set being less than x (which is stated in the first part), why does the second part require me to account for other possible individuals from the population set which satisfy the X(pi) <= x?

Answer 1

In example 5.4.2 you are sampling from a population $ \prod $ given in example 5.4.1 and $\pi_1,\pi_2,\ldots,\pi_{20}$ are the $20$ members of your sample. Since, there are $20$ members in the population, and the sample you are considering is also of size $20$, the members of every sample will be the same. The difference between samples will be in the order of the sample members.

Note that $X(\pi_i)$ is a Random Variable indicating the fertility of the $i_{th}$ plot in a given sample.

It is equally likely for any of the $20$ plots to be the first, second, or even the tenth plot in a sample. So for a sample that we know nothing about,

$$P(X(\pi_i)\leq x)= F_X(x) \ for \ all \ i=1,2,\ldots,20 $$

where $F_X(x)$ is the CDF of the population $\prod$.

In terms of the first part of the equation, is it given that we don't know what the value of the first individual of the population set is?

We know the value of the first individual of the population, but that order need not be preserved in a sample from the population.

From my understanding, it's the number of members in the set such that every member satisfies the condition of being less than "x". If this is true, I am confused as to how solving for the general population gives an answer for the probability of the fertility of the FIRST member being less than "x"? Thanks! –

Your understanding is correct. I can also see where the confusion creeps in. Let's take a simpler example to demonstrate why the two probabilities will always turn out to be the same. Consider the set $A=\{1,2,3\}$ from which we are taking a sample of size $3$. The $3!=6$ possible samples are $S_1=\{1,2,3\}, S_2=\{1,3,2\}, S_3=\{2,1,3\}, S_4=\{2,3,2\}, S_5=\{3,1,2\}, S_6=\{3,2,1\}$. Then, $$P(First\ element\ of\ sample=1)=$$ $$P(First\ element\ of\ sample=2)=$$ $$P(First\ element\ of\ sample=3)=$$ $$\frac{2}{6}=\frac{1}{3}$$

Note that this fraction is the same as $\frac{1}{number\ of\ elements\ in\ population}$. In plain language, the idea being conveyed is that "choosing the first element of the sample" is equivalent to "choosing an arbitrary element from your population". Therefore, solving for the general population gives a valid answer as it perfectly simulates the underlying situation.

PS: I like to think of it as an elegant result of Probability that you get used to with practice.