- #1
madilyn
- 13
- 0
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!
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!