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Category Archives: Bernoulli Trials
Point of increase
Suppose \(U_1,U_2, …\) are independent uniform \( (0,1) \) random variables. Let \(N\) be the first point of increase. That is to say the first \(n \geq 2\) such that \(U_n > U_{n-1}\). Show that for \(u \in (0,1)\):
- \[\mathbf{P}(U_1 \leq u \ { and } \ N=n)= \frac{u^{n-1}}{(n-1)!}-\frac{u^{n}}{n!} \quad;\quad n \geq 2\]
- \( \mathbf{E}(N)=e \)
Some useful observations:
- \[\mathbf{P}(U_1 \leq u \ { and } \ N=n) = \mathbf{P}(U_1 \leq u \ { and } \ N \geq n) -\mathbf{P}(U_1 \leq u \ { and } \ N \geq n+1)\]
- The following events are equal
\[ \{U_1 \leq u \quad{ and } \quad N \geq n\} = \{U_{n-1}\leq U_{n-2} \leq \cdots \leq U_2\leq U_{1} \leq u \}\] - \[ \mathbf{P}\{U_2 \leq U_1 \leq u \}= \int_0^u \int_0^{u_1} du_2 du_1 \]
Blocks of Bernoulli Trials
In \(n+m\) independent \(\text{Bernoulli}(p)\) trials, let \(S_n\) be the number of successes in the first \(n\) trials and \(T_m\) number of successes in the last \(m\) trials.
- What is the distribution of \(S_n\) ? Why ?
- What is the distribution of \(T_m\) ? Why ?
- What is the distribution of \(S_n+T_m\) ? Why ?
- Are \(S_n\) and \(T_m\) independent ? Why ?
[Pitman p. 159, # 10]
Airline Overbooking
An airline knows that over the long run, 90% of passengers who reserve seats for a flight show up. On a particular flight with 300 seats, the airline sold 324 reservations.
- Assuming that passengers show up independently of each other, what is the chance that the flight will be overbooked ?
- Suppose that people tend to travel in groups. Would that increase of decrease the probability of overbooking ? Explain your answer.
- Redo the the calculation in the first question assuming that passengers always travel in pairs. Are your answers to all three questions consistent ?
[Pitman p. 109, #9]
Biased coins
Given a random variable \(x = \{ 0,1\} \) where \(0\) corresponds to heads and \(1\) corresponds to tails.
For a single coin flip: \( \mathbf{P}(x \mid p) = p^x(1-p)^{1-x}\).
For a sequence of \(n\) coin flips: \( \mathbf{P}(x_1,…,x_n \mid p) = \prod_{i=1}^n p^{x_i}(1-p)^{1-x_{i}}\).
I have a bag with three types of coins with the following probabilities of drawing each type:
\( \mathbf{P}(p=.5) = .7 \), \( \mathbf{P}(p=.1) = .2 \), \( \mathbf{P}(p=.9) = .1 \).
I draw a coin from the bag. I flip it \(n\) times resulting in a sequence \(X_1,…,X_n\).
- Using Bayes rule provide the formula for
\[ \mathbf{P}(p = .1 \mid x_1,…,x_n),\quad
\mathbf{P}(p = .5 \mid x_1,…,x_n), \quad\text{and}\quad
\mathbf{P}(p = .9 \mid x_1,…,x_n) \]. - If \(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)=(1,1,0,1,1,0,0,1)\) what is the most likely value probability \(p \) of the coin the was used ?
Blocks of Bernoulli Trials
In \(n+m\) independent Bernoulli \((p)\) trials, let \(S_n\) be the number of successes in the first \(n\) trials, \(T_n\) the number of successes in the last \(m\) trials.
- What is the distribution of \(S_n\) ? Why ?
- What is the distribution of \(T_m\) ? Why ?
- What is the distribution of \(S_n+T_m\) ? Why ?
- Are \(S_n\) and \(T_m\) independent ? Why ?
- Are \(S_n\) and \(T_{m+1}\) independent ? Why ?
- Are \(S_{n+1}\) and \(T_{m}\) independent ? Why ?
Based on [Pitman, p. 159, #10]
