Let $X$ be a random variable distributed over, for example say, the Binomial Distribution. Then $P(X)$ is the probability of getting $x$ successful trials in $n$ total trials.
So I saw a notation that represents the mean of random variables that made me I feel sceptical about my understanding of all the notations I have known. So here's my understanding of the notations:
When it says the expectation of $X$, $E(X)$, does it mean over a long
run, $E(X)$ is the likely number of successful trials? In other words,
the expected value of $X$ is the expected number of successful trials
we would expect in a long run?When it says the variance of $X$, $Var(X)$, does it mean how spread
out the probability of successful trials are? Like how far apart the
probability between the successful trials are?
Now, here's the confusing part. I see a notation like this: $\overline { X } =\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { X }_{ i } } $ and this is called the mean of all the random variables. But it doesn't seem to make sense to me. $X$ is the random variable and carries the value that is the number of successful trials. The average of $X$ is like the average number of successful trials?
Does it then mean $\overline { X } =E(X)$?
Then, there is also the expectation of the mean of all the random variables, $E(\overline { X } )$. So does this represent the average of the average of all the random variables, which means $E(\overline { X } )=E(E(X))$? But at this point, I couldn't understand what it means intuitively. What does it mean here to say the average of the average of all random variables?
Similarly, $Var(\overline { X } )$ is also a confusing term to me. Since $\overline { X } $ is just the average value, what spread does it have?
What is the intuitive meaning of this $\overline { X } $ mean of all random variables $X$ and what does this add on to the meaning of $E(\overline { X } )$ and $Var(\overline { X } )$?
Best Answer
You have a vast population of people who have different heights, and you choose one at random. That person's height is $X$. $E(X)$ is the average height of everyone in the population. Then you pick $20$ people at random. Their heights are $X_1,\ldots,X_{20}$. Their average height is $\bar{X} = (X_1+\cdots+X_{20})/20$. That is a random variable because if you pick another set of $20$, it has a different value---thus it varies randomly. The expected value $E(\bar{X})$ is the same as the expected value $E(X)$. But the variance $\operatorname{var}(\bar{X})$ is smaller than the variance $\operatorname{var}(X)$, because on average one sample of $20$ differs less from another sample of $20$ than one individual differs from another individual.