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Conditioning a Poisson Arrival Process

Consider a Poisson process with parameter  \(\lambda\). What is the conditional probability that \(N(1) = n\) given that \(N(3) = n\)? (Here, \(N(t) \) is the number of calls which arrive between time 0 and time \(t\). ) Do you understand why this probability does not depend on \(\lambda\)?

 

[Meester ex 7.5.4]

Calls arriving

Assume that calls arrive at a call centre according to a Poisson arrival process  with a rate of  15 calls per hour. For \(0 \leq s < t\), let \(N(s,t)\) denote the number of calls which arrive between time \(s\) and \(t\) where time is measured in hours.

  1. What is \( \mathbf{E}\big(\,N(3,5)\,\big)\) ?
  2. What is the second moment of \(N(2,4) \) ?
  3. What is \( \mathbf{E}\big(\,N(1,4)\,N(2,6)\,\big)\) ?

Approximating sums of uniform random variables

Suppose \(X_1,X_2,X_3,X_4\) are independent uniform \((0,1)\) and we set \(S_4=X_1+X_2+X_3+X_4\). Use the normal approximation to estimate \(\mathbf{P}( S_4 \geq 3) \).

geometric probability: marginal densities

Find the density of the random variable \(X\) when the pair \( (X,Y) \) is chosen uniformly from the specified region in the plane in each case below.

  1. The diamond with vertices at \( (0,2), (-2,0), (0,-2), (2,0) \).
  2. The triangle with vertices \( (-2,0), (1,0), (0,2) \).

[Pitman p 277, #12]

probability density example

Suppose  \(X\) takes values in\( (0,1) \) and has a density

\[f(x)=\begin{cases}c x^2 (1-x)^2 \qquad &x\in(0,1)\\  0 & x \not \in (0,1)\end{cases}\]

for some \(c>0\).

  1. Find \( c \).
  2. Find \(\mathbf{E}(X)\).
  3. Find \(\mathrm{Var}(X) \).

 

Games with Black and White Balls

Consider the following gambling game for two players, Black and White. Black puts \(b\) black balls and White puts \(w\) white balls in a box. Black and White take turns at drawing randomly from the box, with replacement between draws until either Black wins by drawing a black ball or White wins by drawing a white ball. Suppose Black gets to draw first.

  1. Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
  2. What value of \(p\) makes the game fair (equal chances of wining) ?
  3. Is the game ever fair ?
  4. What is the least total number of balls in the game, \((b+w)\), such that neither player has more that that \(51\%\) chance of winning ?

 

[Pitman P219, #13]

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