STANDARD REPRESENTATION OF MULTIVARIATE FUNCTIONS
... exposition. Consider for simplicity a single function f : Ω2 → S. It is obvious that the random variables f (Xi , Xj ), i 6= j, form an exchangeable family. Conversely, the theorem by Aldous–Hoover [10, Theorem 7.22] says that every exchangeable family can be represented on ([0, 1], B, dx), but in t ...
... exposition. Consider for simplicity a single function f : Ω2 → S. It is obvious that the random variables f (Xi , Xj ), i 6= j, form an exchangeable family. Conversely, the theorem by Aldous–Hoover [10, Theorem 7.22] says that every exchangeable family can be represented on ([0, 1], B, dx), but in t ...
Multiple Choice Questions
... variable with mean µ = 10 ounces and standard deviation = 1 ounce. 9. Suppose we pick two oranges at random from the bin. The difference in the weights of the two oranges selected (the weight of the first orange minus the weight of the second orange) is a random variable with a mean (in ounces) of ...
... variable with mean µ = 10 ounces and standard deviation = 1 ounce. 9. Suppose we pick two oranges at random from the bin. The difference in the weights of the two oranges selected (the weight of the first orange minus the weight of the second orange) is a random variable with a mean (in ounces) of ...
Available - The University of Texas at Dallas
... 1. Travelbyus is an internet-based travel agency wherein the customers can see videos of the cities they plan to visit. The number of hits daily is normally distributed with a mean of 1,000 and a standard deviation of 240. (a) What is the probability of getting fewer than 900 hits? (b) The website o ...
... 1. Travelbyus is an internet-based travel agency wherein the customers can see videos of the cities they plan to visit. The number of hits daily is normally distributed with a mean of 1,000 and a standard deviation of 240. (a) What is the probability of getting fewer than 900 hits? (b) The website o ...
TPS4e_Ch5_5.1
... State: What is the question of interest about some chance process? Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes of the chance process and what variable to measure. Do: Perform many repetitions of the simulation. Conc ...
... State: What is the question of interest about some chance process? Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes of the chance process and what variable to measure. Do: Perform many repetitions of the simulation. Conc ...
The probability of an event, expressed as P(event), is always a
... 2) You select a person at random from a large conference group. What’s the probability that the person has a birthday in July? Assume 365 days in a year. 3) What’s the probability that a family with 3 children has 2 boys and 1 girl? ...
... 2) You select a person at random from a large conference group. What’s the probability that the person has a birthday in July? Assume 365 days in a year. 3) What’s the probability that a family with 3 children has 2 boys and 1 girl? ...
Bayes for Beginners - Wellcome Trust Centre for Neuroimaging
... What are the priors? • In “classical” SPM, no (flat) priors • In “full” Bayes, priors might be from theoretical arguments or from independent data • In “empirical” Bayes, priors derive from the same data, assuming a hierarchical model for generation of the data Parameters of one level can be made pr ...
... What are the priors? • In “classical” SPM, no (flat) priors • In “full” Bayes, priors might be from theoretical arguments or from independent data • In “empirical” Bayes, priors derive from the same data, assuming a hierarchical model for generation of the data Parameters of one level can be made pr ...
Typical Test Problems (with solutions)
... The conventional solution is p= C4,2C6,2/C6,4=0.429. Trying to be silly, we can also use a Binomial formula assuming that the probabilities are 3/5 and 2/5: p = C4,2(2/5)2(3/5)2= 0.346. This time the difference is significant. In addition, we can now understand the source of the problem, The Binomi ...
... The conventional solution is p= C4,2C6,2/C6,4=0.429. Trying to be silly, we can also use a Binomial formula assuming that the probabilities are 3/5 and 2/5: p = C4,2(2/5)2(3/5)2= 0.346. This time the difference is significant. In addition, we can now understand the source of the problem, The Binomi ...
Problems with computational methods in population
... For appropriate choice of Q this estimator will have much smaller variance than (1). Indeed, if Q is the posterior distribution P (H j An) then the estimator (2) will have zero variance, but unfortunately this posterior distribution is not known for most cases of interest. In any case, it is desira ...
... For appropriate choice of Q this estimator will have much smaller variance than (1). Indeed, if Q is the posterior distribution P (H j An) then the estimator (2) will have zero variance, but unfortunately this posterior distribution is not known for most cases of interest. In any case, it is desira ...
Methods of Assigning Probability
... The classical method for assigning probability, even though being convenient and having welldeveloped mathematical ground, cannot be applied to many reallife statistical problems because the above conditions cannot be satisfied. 2. Relative frequency method of assigning probabilities When the ...
... The classical method for assigning probability, even though being convenient and having welldeveloped mathematical ground, cannot be applied to many reallife statistical problems because the above conditions cannot be satisfied. 2. Relative frequency method of assigning probabilities When the ...
Massachusetts Institute of Technology (Spring 2006)
... she will not lecture unless at least k of the n students enrolled in the class are present. Each student will independently show up with probability pg if the weather is good, and with probability pb if the weather is bad. Given the probability of bad weather on a given day, calculate the probabilit ...
... she will not lecture unless at least k of the n students enrolled in the class are present. Each student will independently show up with probability pg if the weather is good, and with probability pb if the weather is bad. Given the probability of bad weather on a given day, calculate the probabilit ...
Solution - University of Arizona Math
... standard normal pdf. Let Z = Y − X + 4. (a) Find the mean and variance of Z. Solution: E[Z] = E[Y ] − E[X] + 4 = 4. var(Z) = var(Y ) + var(−X) = var(Y ) + var(X) = 1 + 1 = 2. (b) Find the probability density function (pdf) of Z. Hint: this can be done with very little computation. Solution: It is ea ...
... standard normal pdf. Let Z = Y − X + 4. (a) Find the mean and variance of Z. Solution: E[Z] = E[Y ] − E[X] + 4 = 4. var(Z) = var(Y ) + var(−X) = var(Y ) + var(X) = 1 + 1 = 2. (b) Find the probability density function (pdf) of Z. Hint: this can be done with very little computation. Solution: It is ea ...
Probability box
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A probability box (or p-box) is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes.An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds may have almost any shapes, including step functions, so long as they are monotonically increasing and do not cross each other. A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.