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Category Archives: Basic probability
Independence of two hearts ?
Consider a deck of 52 cards. Let \(A\) be the event that the first card is a heart. Let \(B\) be the event that the 51st card is a heart.
What is \(\mathbf{P}(A)\) ? What is \(\mathbf{P}(B)\) ? Are \(A\) and \(B\) independent ?
Conditionally equally Likely
Let \(A\) and \(B\) be two events with positive probability. When does \(\mathbf{P}(A|B)=\mathbf{P}(B|A)\) ?
Making a biased coin fair
Jack and Jill want to use a coin to decide who gets the remaining piece of cake. However, since the coin is Jack’s, Jill is suspicious that the coin is a trick coin which produced head with a probability \(p\) which is not \(\frac12\). Can you devise a way to use this coin to come to a fair decision as to who gets the cake ?
conditional densities
Let \(X\) and \(Y\) have the following joint density:
\[ f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}\]
- Find the marginal densities of \(X\) and \(Y\)
- find \(f_{Y|X}( y \,|\, X=\frac14)\)
- find \( \mathbf{E}(Y \,|\, X=\frac14)\)
[Pitman p426 # 2]
Expected max/min given min/max
Let \(X_1\) and \(X_2\) be the numbers on two independent fair-die rolls. Let \(M\) be the maximum and \(N\) the minimum of \(X_1\) and \(X_2\). Calculate:
- \(\mathbf{E}( M| N=x) \)
- \(\mathbf{E}( N| M=x) \)
- \(\mathbf{E}( M| N) \)
- \(\mathbf{E}( N| M) \)
Difference between max and min
Let \(U_1,U_2,U_3,U_4,U_5\) be independent, each with uiform distribution on \((0,1)\). Let \(R\) be the distance between the max and the min of the \(U_i\)’s. Find
- \(\mathbf{E} R\)
- the joint density of the max and the min of the \(U_i\)’s.
- the \(\mathbf{P}(R> .5)\)
[pitman p355, #14]
A Joint density example II
If \(X\) and \(Y\) have joint density function
\[f(x,y)=\frac{1}{x^2y^2} \quad; \quad x \geq 1, y\geq 1\]
- Compute the joint density fiction of \(U=XY\), \(V=X/Y\).
- What are the marginal densities of \(U\) and \(V\) ?
[Ross p295, # 54]
Closest Point
Consider a Poisson random scatter of points in a plane with mean intensity \(\lambda\) per unit area. Let \(R\) be the distance from zero to the closest point of the scatter.
- Find a formula for the c.d.f. and the density of \(R\) and sketch their graphs.
- Show that \(\sqrt{2 \lambda \pi} R\) has the Rayleigh distribution.
- Find the mean and mode of \(R\).
[pitman p 389, # 21]
Joint Density of Arrival Times
Let \(T_1 < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?
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
\[ \{U_1 \leq u \quad{ and } \quad N \geq n\} = \{U_{n-1}\leq U_{n-2} \leq \cdots \leq U_2\leq U_{1} \leq u \}\] - \[ \mathbf{P}\{U_2 \leq U_1 \leq u \}= \int_0^u \int_0^{u_1} du_2 du_1 \]
An example of min and change of variable
Suppose \(R_1\) and \(R_2\) are two independent random variables with the same density function
\[f(x)=x\exp(-{\textstyle \frac12 }x^2)\]
for \(x\geq 0\). Find
- the density of \(Y=\min(R_1,R_2)\);
- the density of \(Y^2\)
[Pitman p. 336 #21]
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 \( \mathbf{E}\big(\,N(1,4)\,N(2,6)\,\big)\) ?
Tail-sum formula for continuous random variable
Let \(X\) be a positive random variable with c.d.f \(F\).
- Show using the representation \(X=F^{-1}(U)\) where \(U\) is \(\textrm{unif}(1,0)\) that \(\mathbf{E}(X)\) can be interpreted as the area above the graph on \(y=F(x)\) but below the line \(y=1\). Using this deduce that
\[\mathbf{E}(X)=\int_0^\infty [1-F(x)] dx = \int_0^\infty \mathbf{P}(X> x) dx \ .\] - Deduce that if \(X\) has possible values \(0,1,2,\dots\) , then
\[\mathbf{E}(X)=\sum_{k=1}^\infty \mathbf{P}(X\geq k)\]
Min, Max, and Exponential
Let \(X_1\) and \(X_2\) be random variables and let \(M=\mathrm{max}(X_1,X_2)\) and \(N=\mathrm{min}(X_1,X_2)\).
- Argue that the event \(\{ M \leq x\}\) is the same as the event \(\{X_1 \leq x, X_2 \leq x\}\) and similarly that t the event \(\{ N > x\}\) is the same as the event \(\{X_1 > x, X_2 > x\}\).
- Now assume that the \(X_1\) and \(X_2\) are independent and distributed with c.d.f. \(F_1(x)\) and \(F_2(x)\) respectively . Find the c.d.f. of \(M\) and the c.d.f. of \(N\) using the proceeding observation.
- Now assume that \(X_1\) and \(X_2\) are independently and exponentially distributed with parameters \(\lambda_1\) and \(\lambda_2\) respectively. Show that \(N\) is distributed exponentially and identify the parameter in the exponential distribution of \(N\).
- The route to a certain remote island contains 4 bridges. If the time to collapse of each bridge is exponential distributed with mean 20 years and is independent of the other bridges, what is the distribution of the time until the road is impassable because one of the bridges has collapsed.
[Jonathan Mattingly]
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 \(\mathrm{Var}(X) \).
Infinite Mean
Suppose that \(X\) is a random variable whose density is
\[f(x)=\frac{1}{2(1+|x|)^2} \quad x \in (-\infty,\infty)\]
- Draw a graph of \(f(x)\).
- Find \(\mathbf{P}(-1 <X<2)\).
- Find \(\mathbf{P}(X>1)\).
- Is \(\mathbf{E}(X) \) defined ? Explain.
Raindrops are falling
Raindrops are falling at an average rate of 30 drops per square inch per minute.
- What is the chance that a particular square inch is not hit by any drops during a given 10-second period ?
- If one draws a circle of radius 2 inches on the ground, what is the chance that 4 or more drops hits inside the circle over a two-minute period?
- If each drop is a big drop with probability 2/3 and a small drop with probability 1/3, independent of the other drops, what is the chance that during 10 seconds a particular square inch gets hit by precisely four big drops and five small ones?
[Pitman p. 236, #17, Modified by Mattingly]
Overloading an Elevator
A new elevator in a large hotel is designed to carry about 30 people, with a total weight of up to 5000 lbs. More that 5000 lbs. overloads the elevator. The average weight of guests at the hotel is 150 lbs., with a standard deviation of 55 lbs. Suppose 30 of the hotel’s guests get into the elevator . Assuming the weights of the guests are independent random variables, what is the chance of overloading the elevator ? Give your approximate answer as a decimal.
[Pitman p 204, # 19]
Indicator Functions and Expectations – II
Let \(A\) and \(B\) be two events and let \(\mathbf{1}_A\) and \(\mathbf{1}_B\) be the associated indicator functions. Answer the following questions in terms of \(\mathbf{P}(A)\), \(\mathbf{P}(B)\), \(\mathbf{P}(B \cup A)\) and \(\mathbf{P}(B \cap A)\).
- Describe the distribution of \( \mathbf{1}_A\).
- What is \(\mathbf{E} \mathbf{1}_A\) ?
- Describe the distribution of \(\mathbf{1}_A \mathbf{1}_B\).
- What is \(\mathbf{E}(\mathbf{1}_A \mathbf{1}_B)\) ?
The indicator function of an event \(A\) is the random variable which has range \(\{0,1\}\) such that
\[ \mathbf{1}_A(x) = \begin{cases} 1 &; \text{if $x \in A$}\\ 0 &; \text{if $x \not \in A$} \end{cases}\]
Ordered Random Variables
Suppose \(X\) and \(Y\) are two random variable such that \(X \geq Y\).
- For a fixed number \(T\), which would be greater, \(\mathbf{P}(X \leq T) \) or \(\mathbf{P}(Y \leq T) \).
- What if \(T\) is a random variable ? (If it helps you think about the problem, assume \(T\) takes values in \(\{1,\cdots,n\}\). )
Coin tosses: independence and sums
A fair coin is tossed three times. Let \(X\) be the number of heads on the first two tosses, \(Y\) the number of heads on the last two tosses.
- Make a table showing the joint distribution of \(X\) and \(Y\).
- Are \(X\) and \(Y\) independent ?
- Find the distribution of \(X+Y\) ?
Defective Machines
Suppose that the probability that an item produced by a certain machine will be defective is 0.12.
- Find the probability (exactly) that a sample of 10 items will contain at most 1 defective item.
- Use the Poisson to approximate the preceding probability. Compare your two answers.
[Inspired Ross, p. 151, example 7b ]
Boxes without toys
A cereal company advertises a prize in every box of its cereal. In fact, only about 95% of the boxes have a prize in them. If a family buys one box of this cereal every week for a year, estimate the chance that they will collect more than 45 prizes. What assumptions are you making ?
[Pitman p122, # 9]
Picking a box then a ball
Suppose that there are two boxes, labeled odd and even. The odd box contains three balls numbered 1,3,5 and the even box contains two balls labeled 2,4. One of the boxes is picked randomly by tossing a fair coin.
- What is the probability that a 3 is chosen ?
- What is the probability a number less than or equal to 2 is chosen ?
- The above procedure produces a distribution on \(\{1,2,3,4,5\}\) how does it compare to picking a number uniformly (with equal probability) ?
[Pitman p 37, example 5]
Finding a good phone
At the London station there are three pay phones which accept 20p coins. one never works, another works, while the third works with probability 1/2. On my way to London for the day, I wish to identify the reliable phone, so that I can use it on my return. The station is empty and I have just three 20p coins. I try one phone and it doesn’t work. I try another twice in succession and it works both times. What is the probability that this second phone is the reliable one ?
[Suhov and Kelbert, p.10, problem 1.9]
Meeting in a Tournament
A tennis tournament is organized for \(2^n\) players where each round is single elimination with \(n\) rounds. Two players are chosen at random.
- What is the chance that they meet in the first round or second round ?
- What is the chance they meet in the final or semi-final ?
- What is the chance they do not meet at all ?
[Sudov and Kelbert, p4 problem 1.2]
Betting with Coin Flips
Alice and Bob flip a coin repeatedly. Each time there is a head bob gets a dollar and each time there is a tail Alice gets a dollar.
- What is the probability that Bob and Alice have exactly the same amount of money after \(2n\) flips ?
- What is the chance that Alice has more money after \(2n+1\) flips ?
Chance of Testing Positive
In a certain population of people 5% have a disease. Bob’s road side clinic use a test for the disease which has a 97% of (correctly) returning positive if one has the disease and a 25% chance of (incorrectly) returning a positive if one doesn’t have the disease. If a random person is given the test, what is the chance that the result is positive ?
Now let \(\alpha\) be the chance the test returns a positive if one doesn’t have the disease. (Leave the chance that the test correctly returns a positive is one has the disease at 97%). For what value of \(\alpha\) is the chance the test is correct equal to 5% for a randomly chosen person ?
