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Category Archives: Limit Theorems
Binomial with a random parameter
Let \(X\) be binomial with parameters \(n\), which is constant, and \(\Theta\) which is distributed uniformly on \( (0,1)\).
- Find \(\mathbf{E}(s^X | \Theta)\) for any \(s\).
- Show that for any \(s\)
\[ \mathbf{E} ( s^X) = \frac{1}{n+1} \big(\frac{1-s^{n+1}}{1-s} \big)\]
use this to conclude that \(X\) is distributed uniformly on the set \(\{0,1,2, \cdots, n\}\)
Limit for mixtures
Consider the following mixture distribution.
- Draw \(X \sim \mbox{Be}(p=.3)\)
- If \(X=1\) then \(Y \sim \mbox{Geo}(p_1)\)
- If \(X= 0\) then \(Y \sim \mbox{Bin}(n,p_2)\)
Consider the sequence of random variables \(Y_1,…,Y_{200}\) drawn iid from the above random experiment.
Use the central limit theorem to state the distribution of \(S = \frac{1}{200} \sum_i^{200} Y_i\).
(Here \(\mbox{Be}(p)\) is the Bernoulli distribution with parameter \(p\) and \(\mbox{Geo}(p)\) is the geometric distribution with the parameter \(p\). )
Basic Random Walk
Consider the following “game”: A marker is placed on the real line at the point zero. On each turn a coin is flip which a 1 printed on one side and a -1 printed on the other. If the 1 side lands face up, the marker is moved on unit in the positive direction while if the -1 lands heads up then the marker is moved one unit in the negative direction. If the coin has a probability of \(p\) of landing with the 1 side face up, answer the following questions:
- Let \(p=\frac12\). After 10000 turns if you had to pick one site to find the marker which would you choose ?
- Again let \(p=\frac12\). What is the approximate chance that the marker is further then 100 units from this most likely point after 10000 turns ? What is the approximate chance that the marker is further then 300 units from this most likely point after 10000 turns ?
- Repeat the above questions with \(p=\frac{9}{10}\).