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Category Archives: Basic probability
Strontium
Assume we have a large number of particles \(N\) of Strontium. The decay model for Strontium is exponential in that \(\mathbf{P}(T > t) = e^{- \lambda t}\), this states the probability of a an atom surviving until time \(T\).
- The half-life of a substance is the amount of time it takes for an appreciable amount of the substance to be reduced in half. If the half life of strontium is 28 years what is the decay parameter of the exponential ?
- What is the probability Strontium lasts at least 50 years, \(\mathbf{P}(T > 50) \) ?
- Suppose we have \(5\) radioactive substances, the decay of each of which can be modeled by five exponential random variables \(X_1,…,X_5\) with parameters \(\lambda_1,…,\lambda_5\). Assume the five distributions are independent. What is the pdf for \(\min\{X_1,…,X_5\}\).
Expectation of hierachical model
Consider the following hierarchical random variable
- \(\lambda \sim \mbox{Geometric}(p)\)
- \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
Expectation of geometric
Use the expectation as tail sum tool to compute the expectation of the geometric distribution.
Expectation of mixture distribution
Consider the following mixture distribution.
- Draw \(X \sim \mbox{Ber}(p=.3)\)
- If \(X=1\) then \(Y \sim \mbox{Geometric}(p_1)\)
- If \(X= 0\) then \(Y \sim \mbox{Bin}(n,p_2)\)
What is \(\mathbf{E}(Y)\) ?. (*) What is \(\mathbf{E}(Y | X )\) ?.
Maximum of die rolls
Let \(X_1,…,X_5\) be five iid rolls of six sided die. Let \(Z = \mbox{max}\{X_1,…,X_5\}\). Compute \(\mathbf{E}(Z)\).
Sums of Poisson
Agambler bets ten times on events of probability \(1/10\), then twenty times on events with probability \(1/20\), then thirty times on events with probability \(1/30\), then forty times on events with probability \(1/40\). Assuming the vents are independent, what is the approximate distribution of the number of times the gambler wins ? (use Poisson approx. of binomial)
[Pitman 2.5, pg 227]
Polonium data
Look at the following link to the following table summarizing the radioactive decay counts of polonium recorded by Rutherford and Geiger (1910) representing the number of scintillations in 2608 1/8 minute intervals. For example, there were 57 frequencies of zero counts. The counts can be thought of as being approximately Poisson distributed.
- Use the fact that for the Poisson distribution \( \mathbf{E}[X] = \lambda \) to estimate the rate parameter. This is using the methods of moments to estimate a parameter.
- Maximize the likelihood to estimate \( \lambda\).
Random stock brokers
There are \(15\) stock brokers. The returns (in thousands of dollars) on the brokers are modeled
\( X_1,…,X_{15} \stackrel{iid}{\sim} \mbox{N}(0,1).\)
What is the probability that given the above random model at least one broker would bring in greater than $1000 dollars.
2nd Moment of Shifted Random Variables
Let \(X\) be a random variable with \(\mathbf{E}(X)=\mu\) and \(\mathbf{Var}(X)=\sigma^2\). Show that for any constant \(a\)
\[\mathbf{E}\big[(X-a)^2\big]=\sigma^2+(\mu-a)^2\]
Random Digit
Let \(D_i\) be a random digit chosen uniformly from \(\{0,1,2,3,4,5,6,7,8,9\}\). Assume that each of the \(D_i\) are independent.
Let \(X_i\) be the last digit of \(D_i^2\). So if \(D_i=9\) then \(D_i^2=81\) and \(X_i=1\). Define \(\bar X_n\) by
\[\bar X_n = \frac{X_1 + \cdots+X_n}{n}\]
- Predict the value of \(\bar X_n \) when \(n\) is large.
- Find the number \(\epsilon\) such that for \(n=10,000\) the chance that you prediction is off by more than \(\epsilon\) is about 1/200.
- Find approximately the least value of \(n\) such that your prediction of \(\bar X_n\) is correct to within 0.01 with probability at least 0.99 .
- If you just had to predict the first digit of \(\bar X_{100}\), what digit should you choose to maximize your chance of being correct, and what is that chance ?
[Pitman p206, #30]
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]
The Coupon-Collector
Suppose that there are \(N\) different types of coupons. Each box contains one coupon. The type of the coupon is chosen uniformly among \(\{1,2,\cdots,N\}\).
- If we open \(k\) boxes what is the expected number of different coupons thus we will find ? What is the limit of this quantity as \(k \rightarrow \infty\) ?
- Let \(T_k\), for \(k=1,\cdots,N\), be the number of boxes needed to obtain the \(k\)th unique type of coupon. Clearly \(T_1=1\) . For future reference define \(\tau_1=1\) and \(\tau_k=T_k- T_{k-1}\) for \(k=2,3,\cdots\).
- What is the distribution of \(\tau_k\) ?
- What is the expected value of \(T_N\) ? What is it approximately for \(N\) large ?
- What is the variance of \(\mathbf{Var}(T_N) \)?
- Show that \(\mathbf{SD}(T_N) < cn \) from some constant \(c>0\).
- Use Chebychev’s inequality to show that the probability that for large \(N\), \(T_N\) differs from \(N\log(N)\) by at most only a small multiple of \(N\) with high probabilty.
- (***) What is the distribution of \( (T_N- N\log(N) \,)/N\) as \(N\rightarrow \infty\) ? Hint: It is not normal !
Memorylessness and the Geometric Distribution
Let \(X\) be a random variable with range \(\{0,1,2,3,\dots\}\) and distributed geometrical with probability \(p\).
- Show that for every \(n \geq 0\) and \(j\geq 0\),
\[ \mathbf{P}(X-n=j \mid X\geq n) = \mathbf{P}(X=j)\] - If \(X \) is the time to the failure of a machine, then \(\mathbf{P}( X\geq n) \) is the event that the machine has not failed by time \(n\). Why is the above property called Memorylessness ?
- Show that the geometric distribution is the only random variable with range equal to \(\{0,1,2,3,\dots\}\) with this property.
Fishin’ time!
Stark’s Pond contains 10 trout and 5 bluegill fish. Kyle catches a random number of fish (call the number \(X\)), where \(X \sim \text{Unif}(\{1,\ldots,4\})\). Once caught, that fish is removed from the pond and cannot be caught again. Each new fish comes uniformly from the remaining fish.
(a) What is the chance that Kyle catches all trout?
(b) Suppose all the fish that Kyle caught were trout. Given this information, what is the probability that he caught exactly 5 fish?
[Author Mark Huber. Licensed under Creative Commons.]
Introduction to exponential random variables
Let \(\Omega = \{(x,y):0 \leq x,0 \leq y \leq \exp(-x/2)\}\).
(a) What is the area of \(\Omega\)?
(b) Suppose \(U = (U_1,U_2)\) is drawn uniformly from \(\Omega\). Find \(\mathbf{P}(U_1 \leq 2.3)\).
(c) Find \(\mathbf{P}(U_2 \geq 1)\).
(d) For \(a\) an arbitrary positive real number, find \(\mathbf{P}(U_1 \leq a)\).
[Author Mark Huber. Licensed under Creative Commons.]
Time to play some bridge!
A hand in bridge consists of thirteen cards dealt out from a well shuffled deck.
(a) What is the probability that the bridge hand contains exactly 5 hearts?
(b) What is the probability that the bridge hand contains exactly 5 hearts and 5 spades?
(c) What is the probability that the hand contains exactly 5 cards from at least one suit?
[Author Mark Huber. Licensed under Creative Commons.]
Human error is the most common kind
Permanent Memories has three employess who burn Blu-ray discs. Employee 1 has a 0.002 chance of making an error, employee 2 has a 0.001 chance of making an error, and employee 3 has a 0.004 chance of making an error. The employees burn roughly the same number of discs in a day.
(a) What is the probability that a randomly chosen disc has an error on it?
(b) Given that a disc has an error, what is the probability that employee 1 was the culprit?
(c) Given that a disc has an error and employee 3 was on vacation the day it was burned, what is the probability that employee 2 was the culprit?
[Author Mark Huber. Licensed under Creative Commons.]
Drug testing with Bayes’ Rule
A new drug for leukemia works 25% of the time in patients 55 and older, and 50% of the time in patients younger than 55. A test group consists of 17 patients 55 and older and 12 patients younger than 55.
(a) A patient is chosen uniformly at random from the test group, the drug is administered, and it is a success. What is the probability the patient was in the older group?
(b) A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all four patients?
[Author Mark Huber. Licensed under Creative Commons.]
Adding binomials with equal success probability
Suppose \(X \sim \text{Bin}(10,0.2)\) and \(Y \sim \text{Bin}(5,0.2).\)
(a) Say \(X_1,\ldots,X_{10}\) are iid with \(X = X_1 + \cdots + X_{10}\). What distribution for the \(X_i\) makes this statement true?
(b) Say \(Y_1,\ldots,Y_{5}\) are iid with \(Y = Y_1 + \cdots + Y_{5}\). What distribution for the \(Y_i\) makes this statement true?
(c) Write \(X + Y\) in terms of \(X_1,\ldots,X_{10}\) and \(Y_1,\ldots,Y_{5}\).
(d) What is the distribution of \(X + Y\)?
[Author Mark Huber. Licensed under Creative Commons.]
A simple mean calculation
Suppose that \(X \in \{1,2,3\}\) and \(Y = X+ 1\), and \(\mathbf{P}(X = 1) = 0.3, \ \mathbf{P}(X = 2) = 0.5,\ \mathbf{P}(X = 3) = 0.2.\)
(a) Find \(\mathbf{E}(X)\).
(b) Find \(\mathbf{E}(Y)\).
(c) Find \(\mathbf{E}(X + Y)\).
[Author Mark Huber. Licensed under Creative Commons.]
Approximating binomial probabilities with Stirling
Let \(X\) be a binomially distributed random variable with parameters \(n = 1950\) and \(p = 0.342\).
(a) Approximate \(\mathbf{P}(X = 700)\) using Stirling’s approximation to eight significant digits.
(b) Find \(\mathbf{P}(X = 700)\) exactly to eight significant digits using Wolfram Alpha.
[Author Mark Huber. Licensed under Creative Commons.]
Algebras and Conditioning
Consider two draws from a box with replacement contain 1 red ball and 3 blue balls. Let \(X\) be number of red balls. Let \(Y\) be 1 if the two balls are the same color and 0 otherwise. Let \(Z_i\) be the random variable which returns 1 if the \(i\)-th ball is red.
- What is the sample space.
- Write down the algebra of all events on this sample space.
- What is the algebra of events generated by \(X\) ?
- What is the algebra of events generated by \(Y\) ?
- What is the algebra of events generated by \(Z_1\) ?
- What is the algebra of events generated by \(Z_2\) ?
- Which random variables are determined by an another of the random variables. Why ? How is this reflected in the algebras ?
- (*) What pair of random variables are independent ? How is this reflected in the algebras ?
Only Pairwise Independence
Let \(X_1\) and \(X_2\) be two independent tosses of a fair coin. Let \(Y\) be the random variable equal to 1 if exactly one of those coin tosses resulted in heads, and 0 otherwise. For simplicity, let 1 denote heads and 0 tails.
- Write down the joint probability mass function for \((X_1,X_2,Y)\).
- Show that \(X_1\) and \(X_2\) are independent.
- Show that \(X_1\) and \(Y\) are independent.
- Show that \(X_2\) and \(Y\) are independent.
- The three variables are mutually independent if for any \(x_1 \in Range(X_1), x_2\in Range(X_2), y \in Range(Y)\) one has
\[ \mathbf{P}(X_1=x_1,X_2=x_2,Y=y) = \mathbf{P}(X_1=x_1) \mathbf{P}(X_2=x_2) \mathbf{P}(Y=y) \]
Show that \(X_1,X_2,Y\) are not mutually independent.
Expected Value and Mean Error
Let \(X\) be a random variable with \(\mu_1=\mathbf{E}(X)\) and \(\mu_2=\mathbf{E}(X^2)\). For any number \(a\) define the mean squared error
\[J(a)=\mathbf{E}\big[(X-a)^2\big] \]
and the absolute error
\[K(a)=\mathbf{E}\big[|X-a|\big] \]
- Write \(J(a)\) in terms of \(a\), \(\mu_1\), and \(\mu_2\) ?
- Use the above answer to calculate \(\frac{d J(a)}{d\, a}\) .
- Find the \(a\) which is the solution to \(\frac{d J(a)}{d\, a}=0 ?\) Comment on this answer in light of the name “Expected Value” and argue that it is actually a minimum.
- Assume that \(X\) only takes values \(\{x_1,x_2,\dots,x_n\}\). Use the fact that
\[ \frac{d\ }{d a} |x-a| = \begin{cases} -1 & \text{if \(a < x\)}\\
1 & \text{if \(a > x\)}\end{cases}
\]
to show that as long as \(a \not\in \{x_1,x_2,\dots,x_n\}\) one has
\[ \frac{d K(a)}{d\, a} =\mathbf{P}(X<a) – \mathbf{P}(X>a)\] - Now show that if \( a \in (x_k,x_{k+1})\) then \(\mathbf{P}(X<a) – \mathbf{P}(X>a) = 2\mathbf{P}(X \leq x_k) – 1\).
- The median is any point \(a\) so that both \(\mathbf{P}(X\leq a) \geq \frac12 \) and \(\mathbf{P}(X\geq a) \geq\frac12\). Give an example where the median is not unique. (That is to say there is more than one such \(a\).
- Use the above calculations to show that if \(a\) is any median (not equal to one of the \(x_k\)), then it solves \(\frac{d K(a)}{d\, a} =0\) and that it is a minimizer.
Three Valued Random Variable
Show that the distribution of a random variable \(X\) with possible values of 0,1 and 2 is determined by \(\mu_1=\mathbf{E}(X)\) and \(\mu_2=\mathbf{E}(X^2)\), by finding a formula for \(\mathbf{P}(X=x)\) in terms of \(\mu_1\) and \(\mu_2\).
[Pitman p. 184. #20]
Indicator Functions
Let \(A\) and \(B\) be independent events. Let \(\mathbf{1}_A\) and \(\mathbf{1}_B\) be the associated indicator random variables.
- Describe the random variable \(\mathbf{1}_A+ \mathbf{1}_B \) in terms of \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\) ?
- Calculate \(\mathbf{E}(\mathbf{1}_A+ \mathbf{1}_B )\).
- Describe the random variable \((\mathbf{1}_A+ \mathbf{1}_B )^2\) in terms of \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\) ?
- Calculate \(\mathbf{E}\big( (\mathbf{1}_A+ \mathbf{1}_B )^2 \big)\).
[Partially inspired by Pitman p182, #10]
Joint Distributions of Uniforms
Let \(X\) and \(Y\) be independent, each uniformly distributed on \(\{1,2,\dots,n\}\). Find:
- \(\mathbf{P}( X=Y)\)
- \(\mathbf{P}( X < Y)\)
- \(\mathbf{P}( X>Y)\)
- \(\mathbf{P}( \text{max}(X,Y)=k )\) for \(k=1,\dots,n\)
- \(\mathbf{P}( \text{min}(X,Y)=k )\) for \(k=1,\dots,n\)
- \(\mathbf{P}( X+Y=k )\) for \(k=2,\dots,2n\)
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]
Conditional Poisson
The following is a hierarchical model.
- \(\lambda \sim Uniform[1,2]\)
- \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
What is \(\mathbf{E}(Y)\) ?
Mixture of Poisson
The following is a mixture model. The following experiment is used to draw a random variable \(Y\). With probability \(p\) draw from a Poisson distribution with parameter \(\lambda = 1\) so with probability \(1-p\) you are drawing from a Poisson distribution with parameter \(\lambda =2 \).
What is \(\mathbf{E}(Y)\) ?
