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https://en.wikipedia.org/wiki/Expected_value

for a random variable $X$ is given by

$$\mathbb{E}(X)=\sum_{i=1}^{\infty}x_ip_i$$

the $p_i$s are given by the probability distribution (probability function) of $X$.

Are the probability distribution and and the values $x_i$ necessarily linked?

Or

What are $x_i$s in relation to $p_i$s?

mavavilj
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  • That's what I'm asking. What are $x_i$s in relation to $p_i$s. – mavavilj Nov 19 '15 at 20:14
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    The answer by user777 is telling you that the $x_i$'s are the only values for which $P_X$ has a positive mass. Hence if you know $P_X$ you know the support of $P_X$. Which means which values $X$ can take. – Xi'an Nov 19 '15 at 20:16
  • @Xi'an What if I see an $X_i$ in some equation. What's the value of an $X_i$? If I know only $P_X$? E.g. in the mean $\mu=\sum_{i=1}^{N}\frac{1}{N}X_i$. – mavavilj Nov 19 '15 at 21:30
  • The meaning of $X_i$ in some equation would depend on the equation and context. If $X_i$ is the $i$th observation of a random variable, all we know is that it takes some value in the range of $p_X$. If we know $p_X$, then we know the probability that it takes each value. In the expression $\mu=\frac{1}{N}\sum_{i=1}^N X_i$, the expression is denoting $\mu$ as the arithmetic mean of $X$. – Sycorax Nov 19 '15 at 23:45

3 Answers3

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The values $x_i$ are the range of $X$. That is, $x_i\in\{z:p_X(z)>0\},$ where $p_X$ is the p.m.f. of $X$.

Sycorax
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It seems that your confusion is about $X$'s in general. First, you could start with two nice threads that provide definitions and intuitions of random variables. Next, we can try translate it to the plain English. There is some random variable $X$, that has some domain $\Omega$. $\Omega$ is a set of all the possible outcomes of $X$ (all the possible $x$'s). Sometimes $\Omega$ contains only zeros and ones, sometimes it contains only integers, sometimes all the real numbers - it could be anything, but for each random variable the domain is defined.

Next, let's define what $P_X$'s are. $P_X$ is a function that returns numbers from $[0, 1]$ range for each of the possible $x$'s. It can be any function that obeys few axioms defined by Kolmogorov, but this is of less importance in here. $P_X$ function can return the same values of probability for different $x$'s (for example there is $0.5$ probability of throwing head and $0.5$ probability of throwing tail). So it's $P_X$ function that returns probabilities for $x$'s - that is the relation.

The relation between probability $p$ and $x$ such that $P_X(X = x) = p$ exists only in terms of this $P_X$ function and $X$ random variable, while $x$'s are possible values of $X$.

Community
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Tim
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If we have a discrete random variable $X$ on $(\Omega, \mathscr F, \mathbb P)$ that has range $x_1, x_2, ...$, then we have

$$E[X] = \sum_{k=1}^{\infty} x_kP(X=x_k)$$

Hence the $p_k := P(X=x_k)$

BCLC
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