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Simple problems regarding sum of IID random variables

  1. May 30, 2014 #1
    Hi! I'm taking my first course in statistics and am hoping to get some intuition for this set of problems...

    Suppose I have a bowl of marbles that each weighs [itex]m_{marble}=0.01[/itex] kg.

    For each marble I swallow, there is a chance [itex]p=0.53[/itex] that it adds [itex]m_{marble}[/itex] to my weight, and chance [itex]1-p[/itex] that it causes me to puke, therefore losing [itex]m_{puke}=0.011[/itex] kg of my weight.

    1. Assume I religiously swallow [itex]n=10^{4}[/itex] marbles each day. What fraction of the days do I expect to gain weight on?

    Let [itex]X_{i}[/itex]denote the random variable for my weight gained for each swallowed marble, indexed by [itex]i\in\mathbb{Z}^{+}[/itex].

    Let [itex]Y[/itex] denote the random variable for my total weight gained each day from swallowing [itex]n[/itex] marbles, [itex]Y=\sum_{i=1}^{n}X_{i}[/itex]. Then, denote
    [tex]E\left(X\right) := E\left(X_{1}\right)=E\left(X_{2}\right)...=E\left(X_{n}\right)[/tex]
    [tex]Var\left(X\right) := Var\left(X_{1}\right)=Var\left(X_{2}\right)=...=Var\left(X_{n}\right)[/tex]
    such that the theoretical distribution of my daily weight gain is approximately normal with mean
    [tex]E\left(Y\right)=E\left(X_{1}+...+X_{n}\right)=nE\left(X\right)[/tex]
    and variance
    [tex]Var\left(Y\right)=Var\left(X_{1}+...+X_{n}\right)=nVar\left(X\right)[/tex]
    Then, I expect to gain weight on [itex]1-P\left(Y\leq0\right)\approx0.892[/itex] of the days. Is this correct?

    2. Why does [itex]Y[/itex] approximately follow the distribution [itex]N\left(nE\left(X\right),\sqrt{nVar\left(X\right)}\right)[/itex]?

    Firstly, am I correct that the variance is [itex]nVar\left(X\right)[/itex] and not [itex]n^{2}Var\left(X\right)[/itex]? Can someone refresh me what's the intuitive difference between the random variable [itex]Y=X_{1}+...+X_{n}[/itex] as compared to [itex]Y=50X[/itex] again?

    Secondly, it's not immediately obvious to me how the distribution approaches a Gaussian distribution as [itex]n\rightarrow\infty[/itex]? Perhaps I can formulate this in terms of the convolution of a discrete function representing the distribution of my weight gain/loss for each marble swallowed? Will the discrete convolution approach generalize nicely to the sum of any discrete random variable?

    3. For finite [itex]n[/itex], am I correct that this distribution converges faster to a Gaussian distribution near the center and slower in the tails as [itex]n[/itex] increases? Can I quantify this rate of convergence?

    Just from my intuition, I think the best strategy to attack this problem is to express the sum of the random variables [itex]Y=\sum_{i=1}^{n}X_{i}[/itex] as a Fourier transform and then investigate the rate of convergence using an asymptotic expansion of the integral in large [itex]n[/itex] i.e. saddle point method?

    Thanks!
     
  2. jcsd
  3. May 30, 2014 #2

    Stephen Tashi

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    This thread will probably be moved to the homework section. (I think the math homework section would be more obvious as a "math homework" subsection of mathematics - so I sympathize with the misplacement.) For homework, you should state the problem - rather than leave the helpers to guess it from reading your reasoning.


    The question of "did gain" vs "did not gain" is a different question that "how much" is gained. "Did gain" can be represented by a random variable that only takes on the values 1 or 0. (A bernoulli random variable.) The theorems about expectation and variance apply to such a random variable but you'd get different answers than you get from answering questions about "how much".


    Think of a computer simulation. The algorithm for simulating 50 X is to make a random determination for X and then multiply it by 50. The algorithm for simulation the sum of 50 different realizations of X is obviously to make 50 random determinations of X and add them. When you make 50 different random determinations, you have the possibility that opposite extremes will "cancel out". You don't get that with a simulation of 50 X since you only make one random determination for X.


    That's a good question! I'll guess there are several ways, but i don't know them. One thing to do is understand the distinction between "pointwise convergence" and "uniform convergence". Convergence of a sequence of functions to another function is more complicated than convergence of a function evaluated at a sequence of points to a single number. There are several different definitions for "convergence" when we deal with sequences of functions converging to a single function.
     
  4. May 30, 2014 #3
    Oh sorry, I came up with the questions myself so it isn't a homework problem.


    I see. What would you intuitively interpret the 89.2% figure as if not the fraction of days I expect to have positive weight gain?

    Ah, this made a lot of sense! Thanks!

    Yes, I figured this is a difficult problem, but also one of the more interesting ones I've thought of while learning statistics. What's a good starting point for me to learn about the different definitions of "convergence" that you mentioned in your last sentence?

    Thanks so much!
     
  5. May 31, 2014 #4

    FactChecker

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    A couple of notes. (Talking informally in terms of X=sum of events, not divided by n.)

    1) You are talking about approximating a continuous PDF with a discrete PDF so they will always differ significantly at values of X that are not possible for the discrete PDF. You can handle this several ways. I think the best way is to look at convergence of the CDFs instead of the PDF. The CDF of a discrete real random variable is easy to unambiguously extend to all real values.

    2) You know that the discrete CDF is 1.0 for all values of X > n. And n+epsilon is where the normal distribution CDF is farthest from 1.0. I suspect that this is where the greatest difference between the two CDFs is and that it gives the rate of uniform convergence. I can not prove it without some work.
     
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