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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.
- What is \( \mathbf{E}\big(\,N(3,5)\,\big)\) ?
- What is the second moment of \(N(2,4) \) ?
- 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.
- The diamond with vertices at \( (0,2), (-2,0), (0,-2), (2,0) \).
- 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 \(c>0\).
- Find \( c \).
- Find \(\mathbf{E}(X)\).
- 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.
- Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
- What value of \(p\) makes the game fair (equal chances of wining) ?
- Is the game ever fair ?
- 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]