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Suppose X1 is one observation from a population with Beta(θ,1) PDF. Would X1 also have Beta(θ,1) PDF?

ally.williams
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1 Answers1

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As you can learn from the How to Understand the Relationships Among Random Variables, Samples, and Populations? thread,

A population is usually modeled as a set $\mathcal S$ together with a probability measure $\mathbb{P}$ on that set. [...]
A (univariate) random variable $X:\mathcal{S}\to\mathbb{R}$ assigns numbers to the elements of $\mathcal S$. [...]

and sampling is commonly understood in statistics as

[...] Another procedure focuses on a random variable, rather than the population, and views an independent and identically distributed ("iid") sample as a sequence $$X_1, X_2, \ldots, X_n$$ of random variables on $\mathcal S$ (a) that are independent and (b) for which all the $X_i$ have the same distribution.

So a sample of size $n$ is thought as a sequence of random variables

$$X_1, X_2, \ldots, X_n$$

all following the same distribution, "coming from" the same population. The population does not "have" distribution, random variables do. You are mixing different concepts, so I highly recommend the linked thread that discusses it in depth.

Tim
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