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I know that $E[x] = p$ for some Bernoulli random variable X with parameter $p$ where $P(X = 1) = p$ and $P(X = 0) = 1 − p$. Similarly $E[X^2] = p$ and $Var[X] = p − p^2 = p(1 − p)$.

However, I'm having trouble wrapping my head around the following. Let's say $X_i ∼ Bernoulli(p)$. I was told that we can determine the value of $p$ by using some number $t$ of i.i.d. samples $X_1, ... , X_t ∼ Bernoulli(p)$. What is the expected value of the estimate?

  1. My intuition isn't quite clear here. My initial attempt it is, let us say we have a coin that lands on heads with $p$ and tails with $1-p$. Then, we can flip it some $t$ times, and if we do something like $\frac{\text{# of heads}}{\text{total # of slips}}$ or equivalently $X = \frac{1}{t}\sum^{t}_{i=1}X_i$, then we can estimate $p$. And as $t$ gets larger, the stronger our estimate will be. Is this the right intuition here?

Secondly, if my intuition is correct, then what exactly is the expected value and variance of $X$ in terms of $p$ and $t$. I realize this might be an obvious question, but I'm having trouble differentiating it from just $E[x] = p$.

Am I just suppose to be doing $E[X] = E[X_1] + E[X_2] ...$? How do I put it in terms of $p$ and $t$.

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    $\begingroup$ $X$ will have the same expected value as each $X_i$ due to linearity of the expected value. Additionally, the variance of $X$ will be the variance of $X_i$ divided by $t$. So you can see that adding more samples will reduce the variance of the estimate. $\endgroup$ Oct 5 '20 at 22:57
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You apply the Linearity of Expectation.

Since the mean estimator is indeed $\bar X=\tfrac {\sum_{i=1}^tX_t}t$ then the expectation is:-

$$\begin{align}\mathsf E(\bar X) &= \mathsf E\left(\tfrac 1 t\sum_{i=1}^t X_i\right)\\[1ex]&=\tfrac 1t\sum_{i=1}^t\mathsf E(X_i)\end{align}$$

The rest is that these $t$ samples are identically distributed, and for all $t$ samples their expectation is $p$.

So, yeah, $\mathsf E(\bar X)=p$


Similarly for the variance of the mean estimator, we also use independence (and so the variance of the sum equals the sum of variance):- $$\begin{align}\mathsf{Var}(\bar X)&=\mathsf {Var}(\tfrac 1t\sum_{i=1}^t X_i)\\[1ex]&=\sum_{i=1}^t\mathsf{Var}(\tfrac 1{t}X_i)\\[1ex]&~~\vdots\end{align}$$

Recall we have for any scalar $a$ and random variable $Z$, that: $\mathsf {Var}(aZ)=a^2\mathsf{Var}(Z)$

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    $\begingroup$ So for $Var(X)$, I get $\frac{1}{t^2}\sum^{t}_{i=1}{Var(X_i)}$. And we have $Var(X_i) = E({X_i}^2) - E(X_i)^2 = E({X_i}^2) - p^2$. This is where I'm stuck. From an intuitive sense what does it mean to say $E({X_i}^2)$? Expectation of getting $X_i$ AND $X_i$ ? Not sure how to simply that. $\endgroup$
    – Jonathan
    Oct 6 '20 at 15:59
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    $\begingroup$ I know the answer for $E({X_i}^2)$ is $p$. And so the final answer is $\mathsf{Var}(\bar X) = \frac{p(1-p)}{t}$. But unable to intuitively wrap my head around why we are using this value. $\endgroup$
    – Jonathan
    Oct 6 '20 at 16:05
  • $\begingroup$ @Jonathan , You see that $X_i^2$ takes on values entirely determined by the value for $X_i$, so by definition: $$\begin{align}\mathsf E(X_i^2)&=0^2\cdot\mathsf P(X_i=0)+1^2\cdot\mathsf P(X_i=1)\\ &=0+p\end{align}$$ $\endgroup$ Oct 6 '20 at 23:07

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