If we were able to prove that the universe is infinite, wouldn't that statistically prove that there is no other forms of life? I want to begin my explanation using abstract mathematical explanation to repetition possibility by taking independent samples $X_n$ from some continuous probability distribution: https://math.stackexchange.com/q/1739927/
If we applied this same principle and its conclusion and assumed that the universe is homogeneous and were able to prove its infinity someway (along with matter inside it including stars and planets) then, statistically, it would mean that the chance of repetition of life again, at least and in the most conservative approach here, as a form of doppelganger extraterrestrial life would be zero? meaning that we are alone in this universe
 A: No, quite the opposite. If the universe is truly infinite and approximately homogeneous, then I invite you to point in some direction in the sky at night, and if you travel far enough in that direction, you might have to point through several stars and planets and alien organisms to get there, but eventually you might well be pointing at a nearly exact copy of yourself pointing back at you. With infinite repetition things that are possible can become probable and things that are probable can become certain.
Not everything works this way, and it does depend on how “identical” your setups are as you repeat. But the basic math is like this: imagine we model a bridge but we ignore continuous “damage” to it and treat its failure instead as an all-or-nothing thing which has an 0.01% chance of failing in any given day, independent of any other day. One can derive that this bridge will live about  10,000 days or 27 years, plus or minus another 27. It actually in the continuous limit has a 50/50 chance of failing apart after $10000~\ln2$ days or 19 years. So those independent little chances of something happening, under many many repetitions, eventually lead to this very unlikely thing (one in ten thousand) becoming probable at $10000~\ln2$ repetitions and becoming 99.995% likely after a hundred thousand repetitions. Very simple math. And all that the above statement is doing is the same argument, extrapolating an Earth-sized cylinder through space and saying “hey, that this Earth is roughly the way it is (up to your uncertainty) is some unbelievably tiny probability $p$, but let’s chain those cylinders together in the direction you're pointing until we get $10/p$ repetitions and I suppose if you're right that everything is infinite and homogeneous then we'll not run out of universe before $10/p$ repetitions, even for small $p$.”
The “(up to your uncertainty)” is also very important here, because it is what makes the probability finite rather than infinitesimally zero. Mathematicians deal with say real numbers being perfectly specified, so 1.00125267... is very different from 1.00125264... but us physicists have to say “look my experiment has an 0.2% error and I can't resolve the difference between 1 and 1.002, so all of these are approximately 1.000 up to my error bars.”
Actually, something that bugs some people is the idea of “Boltzmann brains.” This is the idea that generating a brain randomly is much easier than generating a whole body and Earth and brain all together, so that if the world really does proceed to a state of maximum entropy, a thermal equilibrium, then there is some sort of recurrence time (in the simplest case a Poincaré recurrence) where the thermal equilibrium randomly generates a tiny bit of brain like yours perceiving the things that you are perceiving but the vast vast majority of times this happens that brain is totally and utterly detached from the real world and vanishes after a moment. And the reason that this application of infinity bugs people is that it means that averaging over all the entities in our universe which are having the conscious experience that we are having of staring at our computer screens on a physics website pondering the universe, almost none of them are actually living on an Earth and almost none of them are going to last longer than the next five seconds and almost none of them are therefore actually seeing anything like the real world, they are just in a very temporary hallucination. And that brings up questions about why we are so sure that we are not.
A: I think you've misunderstood the question and answer that you linked to.
Paraphrasing, that question is, "Given an infinite sequence of real numbers, each one randomly and independently chosen from the interval $[0, 1]$, what is the probability that there is some number which appears multiple times?" The answer is that the probability is zero.
But why is it zero? Why is it essentially impossible that any duplicates will appear?
You seem to have mistakenly concluded that it's because the sequence is infinite. Actually, the infinity of the sequence has nothing to do with the answer to the question. If that asker had asked about a finite sequence instead of an infinite sequence, the answer would have been exactly the same.
The actual reason that the probability of duplicates in an infinite sequence of real numbers is zero is that the probability of any two randomly chosen real numbers being equal is zero. Selecting infinitely many numbers, instead of two, fails to change the situation.
You've also mistakenly assumed that the universe is similar to a sequence of real numbers, and that the existence of life existing elsewhere is similar to two random real numbers being equal. In fact, there are infinitely many arrangements of particles that constitute life, and so the probability of life existing in any given place is greater than zero. If the universe contains an infinite amount of matter, then the probability of life existing elsewhere is probably 1.
A: The cited mathematical theorem in this context simply means that we would not find any two exactly identical "lives", i.e. no Doppelgängers. This however does not mean that we cannot find very similar ones. Moreover, this is correct, only if we take a finite number of samples (the doppelgänger arguments in the other answers imply that you have to travel infinitely far).
Note that this argument can be turned also against the existence of life on Earth - the probability that the life has arises in the exact form that we know now and that we are having this discussion is zero. But there is an infinite number of very close possibilities, and integrating over them gives a finite probability.
A: The reason why many believe that there are life forms on other planets has nothing to do with infinity but everything to do with what we know about how life evolved in earth and the tremendous number of galaxies and hence the even more tremendous number of stars with planets around them. It's impossible to be be firmer about this because of the lack of data. Given that we have now telescopes that are capable now of inferring ecoplanets, with over 3,000 discovered, that data is only likely to improve.
A: To add something to the discussion, let me state that a Doppelgänger of yourself can't be found anywhere in the universe. Max Tegmark claims that in an infinite universe we can find Hubble volumes (even infinitely many) that contain exactly the same configuration of particles (in phase space) as our own Hubble volume. And thus, a Döppelganger will exist somewhere too (even an infinity of them). This overlooks the possibility of interaction at the boundary of these volumes. Someone between us and the edge of a volume sees a different image as her supposed Doppelgänger would see. Which means that no two volumes can be the same.
Your assumption must be the opposite. If the universe is infinite the chance of life elsewhere is one, assuming the universe is the same everywhere, which seems rather plausible.
If the universe is not the same everywhere then one has to make an assumption of how the universe is different, with associated chances of being different, both of which seem impossible to me. Up to the limit of the visible universe, the laws of Nature seem the same as over here. The visible part is a tiny part though of the universe as a whole, so who knows?
A: If the universe is infinite, that would mean that there is a 100% chance that there is extraterritorial. Let's take a portion of the universe. There is an x% chance of having alien life. If we double that then there would be and x2% chance of alien life. If we double it infinte times then we would get x∞% chance... and since any number multiplied by infinity is equal to infinity, there is 100% of a chance that there is alien inhabited planet, somewhere in the universe...
A: The other answer has already shown that this is not the case.
However, it is also true that: "If we take random samples from a continuous distribution, the probability of repetition is $0$"
We can identify the mistakes that make the conclusion drawn from the above statement invalid
1. What is repeating?:
Note that we can only say that the probability of a repeat of the exact same observation is zero - and not the probability of some random observation from a large class of similar samples.
Recall that humanity in particular, and life on Earth in general do not contain all possible life. Rather we are a very particular case of the infinite (in the sense of infinitely many variations within the set) possibilities of life.
Thus, while we may say that the probability of finding a planet identical to Earth with life exactly like us on it elsewhere in the universe is $0$. This does not mean that the probability of finding life itself - intelligent or otherwise - is $0$. Life is like an interval, not like a point.
In other words, we may not find another Omar Adel asking this question on Physics.SE, but we do find other users asking physics questions on the Internet.
2. Probability 0 $\neq$ Impossible:
This may seem counter-intuitive, but it is the case in certain contexts - including this one.
For example, consider a number line or a Cartesian plane. Because there are infinitely many points, the probability of choosing any given point at random is $0$. However, if we were to run a trail, we will choose some point, even though a priori, the probability of choosing it was $0$. (This would also hold in the case of countable infinities, such as choosing a random natural number. However, note that such distributions would not be continuous - though they can be uniform - and are not relevant here).
Thus, in the context of such continuous distributions, probability zero may not imply impossibility. Indeed, the concept of measure is more suitable here.
Thus, the fact that the probability of a repeat is zero, does not mean that the existence of a copy is impossible.
3. The universe:
More general than the other two critiques is the fact that correspondence between the distribution for which the result holds and the universe is far from exact.
Leaving aside the fact that we currently do not think that the universe is infinite, it is easy to imagine infinite-universes that are counter-examples to this.
Imagine an infinite universe that is empty except for 2 (or more) identical particles rotating around their common center of mass. Here, the variety of structures is not infinite, rather there is either empty space or a particle - and both configurations repeat at least twice!
In fact, many qualitative arguments about infinite universes (including the one in CR Drost's excellent answer) tacitly assume some non-trivial properties that need to be fleshed out before we can say anything that is so mathematically precise. After all, what stops an infinite universe from simply being 'tiled' by a uniform grid of identical atoms?
Thus, we need to make many assumptions before the argument even applies to an arbitrary infinite universe.
A: 
“It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.”

- Douglas Adams, The Restaurant at the End of the Universe
A: Absolutely not.  It would only prove the universe was infinite if you could prove that.  But this is moot as the universe is finite.
Also, if it were infinite there would be more chance of life in the places we cannot see or get to. Would certainly not have any bearing on proving there was no life anywhere.
A: The mind when fully flexed has the power to imagine many things that exist but are unseen, and many things that are unseen because they do not exist.  The constructions you chose are imaginary for two reasons.  First, the universe is observed to be heterogeneous and second, the distribution of life is observed to be self-similar in nature on earth allowing that same self-similarity to be reasonably extrapolated from the observed earth to the universe in whole. The normal distribution cannot strictly be applied to self-similar subjects governed by fractal mathematics.  But of course the unscrupulous can use statistics to prove anything they wish because there are lies, damned lies and then statistics.
