Hi! I'm taking my first course in statistics and am hoping to get some intuition for this set of problems...(adsbygoogle = window.adsbygoogle || []).push({});

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Simple problems regarding sum of IID random variables

Loading...

Similar Threads for Simple problems regarding |
---|

I The Halting Problem |

B Problem in Counting - Number of Passwords |

I A seemingly simple problem about probability |

I A simple question about probability theory |

A Advanced Data Fitting - More than Simple Regressions |

**Physics Forums | Science Articles, Homework Help, Discussion**