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Binomial with a random parameter

Let \(X\) be binomial with parameters \(n\), which is constant, and  \(\Theta\) which is  distributed  uniformly on \( (0,1)\).

  1. Find \(\mathbf{E}(s^X | \Theta)\)  for any \(s\).
  2. 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 theorems via generating functions

Use the results on generating functions and limit theorems which can be found here to answer the following questions.

  1. Let \(Y_n\) be uniform on the set  \(\{1,2,3,\cdots,n\}\). Find the moment generating function of \(\frac1n Y_n\) which we will call it \(M_n(t)\).  Then show that as \(n \rightarrow \infty\),
    \[ M_n(t) \rightarrow \frac{e^t -1}{t}\]
    Lastly, identify this limiting moment generating function that of a known random variable. Comment on why this make sense.
  2. Let \(X_n\) be distributed as a binomial with parameters \(n\) and \(p_n=\lambda/n\).   By using the probability generating function for \(X_n\), show that \(X_n\) converges to a Poisson random variable with parameter \(\lambda\) as \(n \rightarrow \infty\).

[Adapted from Stirzaker, p 318]

Limit for mixtures

Consider the following mixture distribution.

  1. Draw \(X \sim \mbox{Be}(p=.3)\)
  2. If \(X=1\) then \(Y \sim \mbox{Geo}(p_1)\)
  3. 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:

  1. Let \(p=\frac12\). After 10000 turns if you had to pick one site to find the marker which would you choose ?
  2. 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 ?
  3. Repeat the above questions with \(p=\frac{9}{10}\).

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