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 Aug 18 awarded Notable Question Sep 24 awarded Autobiographer Jul 2 awarded Curious Mar 10 awarded Popular Question Mar 7 asked Finding a base for a series to sum to a constant Nov 18 comment Markov chains: simple random walk $S_n$ Suppose that you are at origin and start moving away at discrete random increments. What I understand from your statement is that the probability of being at distance $i$ from the origin at the $n$-th stage of your walk given that you have been at distance $i-1$ in the $n-1$-th stage is given as $\frac{p^i}{p^i+q^i}$. Nov 18 awarded Teacher Nov 18 answered Markov chains: simple random walk $S_n$ Nov 18 awarded Commentator Nov 18 comment Calculating expectation of a random variable knowing expectation of a function of it @StephenMontgomery-Smith: Fair enough. Then, I better rephrase my question as what other information about $X$, except its density function, help attain a tighter bound for (or the exact value of) $\mathbb{E}[X]$? Nov 18 comment Calculating expectation of a random variable knowing expectation of a function of it @StephenMontgomery-Smith: Ah, I see. But $X$ is not necessarily a constant random variable. Nov 18 comment Calculating expectation of a random variable knowing expectation of a function of it @DilipSarwate: I just know $\mathbb{E}[a^X]$ for a specific positive constant $a$ and not as a function of $a$. I am kind of familiar with the concept of moment-generating function, but cannot see how it can be exploited to solve this problem. Nov 18 comment Calculating expectation of a random variable knowing expectation of a function of it What do you mean by "constant random variable"? Nov 18 asked Calculating expectation of a random variable knowing expectation of a function of it Oct 28 accepted Calculating expected value of distance in a circle-circle intersection Oct 24 asked Calculating expected value of distance in a circle-circle intersection Oct 24 awarded Scholar Oct 24 accepted A contradiction when calculating the expected value of a discrete random variable Jul 18 asked Notation for element-wise division of vectors Jun 22 comment Segmented area between circles Thanks, @Victor. I do not believe if it is easy at all to formally prove that $A_{22}$ grows with distance to $t$. I think numerical evaluation of the regions (like what you did in your answer) is the best way to approach the problem. Unfortunately, I am not familiar with Mathematica; so, I am trying to implement some code in MATLAB. I will update the post if I come up with more interesting results.