How does the above notation of PGF read in plain English?
Is it $\alpha$ to the power $X$? What does it even mean?
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Yes, it is the expected value of the random variable $\alpha^X$ (read as usual). Here the expectation always exists because $0\le \alpha\le 1$, and you are also given that $X$ is an integer-valued random variable. – StubbornAtom Oct 26 '18 at 09:29
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Start here: https://en.wikipedia.org/wiki/Probability-generating_function
In general, be careful with generating functions. They're very useful for doing theoretic work, but you can't treat them like a normal finite polynomial. They're an infinite series with useful properties, but the "infinite" part of infinite series can often lead people astray who aren't very practiced with them.
To translate, as you asked, into plain English, it says that the Probability Generating function of the random variable $X$ is written as $g_X(\alpha)$ and is equal to the Expected Value of $\alpha^X$, defined for $\alpha$ between 0 and 1.
$\alpha$ is a dummy variable. What is missing is the summation form, given at the link above. The last part of your picture is the beginning of a property or alternate definition for pgfs that only works if $X$ is integer-valued, but isn't part of the first two lines.
As for what the formula means, since $X$ is a random variable, any formula with $X$ in it can have an expected value calculated by the usual formula of summing over every possible result of $X$, multiplying the formula by the probability of that result occurring.
IE, $$g_X(\alpha)=E(\alpha^X) = \sum_{X} p(X)\alpha^X$$ If $X$ only takes on integer values, this becomes a standard power series.
JKreft
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Yes, that is a number, E, times the number, $\alpha$, to the X power. We can write $\alpha^x= e^{ln(\alpha^x}= e^{xln(\alpha)}= \left(e^{ln(\alpha)}\right)^x$
user247327
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