Lecture 2 - Probability theory
... to come up with a more useful, universal law... The practical and theoretical importance of the normal distribution seems to be related to the following properties: 1. Its connection to the arithmetic mean (it is the unique distribution for which the arithmetic mean is the best estimate in the sense ...
... to come up with a more useful, universal law... The practical and theoretical importance of the normal distribution seems to be related to the following properties: 1. Its connection to the arithmetic mean (it is the unique distribution for which the arithmetic mean is the best estimate in the sense ...
An Introduction to Probability
... with physical models that determine the effects that result from various causes — we know how image intensity is determined, for example. The difficulty is that effects could have come from various causes and we would like to know which — for example, is the image dark because the light level is low, or ...
... with physical models that determine the effects that result from various causes — we know how image intensity is determined, for example. The difficulty is that effects could have come from various causes and we would like to know which — for example, is the image dark because the light level is low, or ...
ECS 455: Mobile Communications Call Blocking Probability
... (b) Then, we divide T into n slots. (For example, n = 10, 000.) (c) For each slot, only two cases can happen: 1 arrival or no arrival. So, we generate Bernoulli random variable for each slot with p1 = λ × T /n. (For example, if λ = 5 arrival/hr, then p1 = 0.01.) To do this for n slots, we can use th ...
... (b) Then, we divide T into n slots. (For example, n = 10, 000.) (c) For each slot, only two cases can happen: 1 arrival or no arrival. So, we generate Bernoulli random variable for each slot with p1 = λ × T /n. (For example, if λ = 5 arrival/hr, then p1 = 0.01.) To do this for n slots, we can use th ...
solutions
... three of Kolmogorov’s axioms hold. Let’s take them one by one. Does Axiom 1 hold? The probability of every event is non-negative – we listed all the events above, and their probabilities are explicitly given in the table. Hence Axiom 1 holds. Does Axiom 2 hold? P (S) = P ({a, b}) = 1, again from the ...
... three of Kolmogorov’s axioms hold. Let’s take them one by one. Does Axiom 1 hold? The probability of every event is non-negative – we listed all the events above, and their probabilities are explicitly given in the table. Hence Axiom 1 holds. Does Axiom 2 hold? P (S) = P ({a, b}) = 1, again from the ...
Stochastic Processes
... sequence (An )n∈N ⊆ K such that An ↑ A or An ↓ A as n → +∞, A ∈ K . 1.2.1 Theorem (Monotone class theorem for sets). Let F be an algebra and K a monotone class of sets of Ω such that F ⊆ K . Then σ(F ) ⊆ K . For the formulation of the monotone class theorem for classes of functions we refer to Sharp ...
... sequence (An )n∈N ⊆ K such that An ↑ A or An ↓ A as n → +∞, A ∈ K . 1.2.1 Theorem (Monotone class theorem for sets). Let F be an algebra and K a monotone class of sets of Ω such that F ⊆ K . Then σ(F ) ⊆ K . For the formulation of the monotone class theorem for classes of functions we refer to Sharp ...
A Poisoned Dart for Conditionals
... conditional. Choose one of these conditionals—say, [½, 1] 2/3. This is true at a world w just in case the closest (overall most similar) world to w in which the dart lands in [½, 1] is a world in which it lands exactly on 2/3. In which worlds is this the case? Consider first the worlds in which th ...
... conditional. Choose one of these conditionals—say, [½, 1] 2/3. This is true at a world w just in case the closest (overall most similar) world to w in which the dart lands in [½, 1] is a world in which it lands exactly on 2/3. In which worlds is this the case? Consider first the worlds in which th ...
7. Discrete probability and the laws of chance
... sense: it means that we have accounted for all possibilities, i.e. the fractions corresponding to all of the outcomes add up to 100% of the results. In a case where there are M possible outcomes, all with equal probability, it follows that pi = 1/M for every i. ...
... sense: it means that we have accounted for all possibilities, i.e. the fractions corresponding to all of the outcomes add up to 100% of the results. In a case where there are M possible outcomes, all with equal probability, it follows that pi = 1/M for every i. ...
3 - Rice University
... Event A: The two dice having the event guessed by Player B Event B: The sum of the two dice being what Player A informed Player B Using the same example of sum = 6 Let us say that Player B guessed as (1,5) P(Event A) = 1/36 = P(Event A AND Event B) For sum = 6 Total number of events = 36 ...
... Event A: The two dice having the event guessed by Player B Event B: The sum of the two dice being what Player A informed Player B Using the same example of sum = 6 Let us say that Player B guessed as (1,5) P(Event A) = 1/36 = P(Event A AND Event B) For sum = 6 Total number of events = 36 ...
DOC - Berkeley Statistics
... probability. The marginal distribution P(Xn) is the distribution over states at time n. The initial distribution is P(X0). There may exist one or more state distributions π such that ...
... probability. The marginal distribution P(Xn) is the distribution over states at time n. The initial distribution is P(X0). There may exist one or more state distributions π such that ...
Ballot theorems for random walks with finite variance
... a simple random walk with step size X. Before stating our ballot theorem, we introduce a small amount of terminology. We say X is a lattice random variable with period d > 0 if there is a constant z such that dX − z is an integer random variable and d is the smallest positive real number for which t ...
... a simple random walk with step size X. Before stating our ballot theorem, we introduce a small amount of terminology. We say X is a lattice random variable with period d > 0 if there is a constant z such that dX − z is an integer random variable and d is the smallest positive real number for which t ...
Probability and Information Theory
... it is not immediately obvious that probability theory can provide all of the tools we want for artificial intelligence applications. Probability theory was originally developed to analyze the frequencies of events. It is easy to see how probability theory can be used to study events like drawing a ce ...
... it is not immediately obvious that probability theory can provide all of the tools we want for artificial intelligence applications. Probability theory was originally developed to analyze the frequencies of events. It is easy to see how probability theory can be used to study events like drawing a ce ...
Conditional Probability
... =1/2. People who receive the morning paper are less likely to receive the evening paper than people who do not receive the morning paper. Since the probability the event E occurs depends on whether or not M occurred, we call these events dependent. There are many events of this form. Any events with ...
... =1/2. People who receive the morning paper are less likely to receive the evening paper than people who do not receive the morning paper. Since the probability the event E occurs depends on whether or not M occurred, we call these events dependent. There are many events of this form. Any events with ...
Probability and Symmetry Paul Bartha Richard Johns
... assignments of equal likelihood or, as we shall say, equi-possibility, to certain outcomes, as well as certain conditional probability assignments. The dart is as likely to hit the left half of the dartboard as the right half; one ticket is as likely as any other to win. Ordinary measures on these s ...
... assignments of equal likelihood or, as we shall say, equi-possibility, to certain outcomes, as well as certain conditional probability assignments. The dart is as likely to hit the left half of the dartboard as the right half; one ticket is as likely as any other to win. Ordinary measures on these s ...
• Elementary propositions can be combined to form complex
... • E.g., P(Weather, Cavity) is a 4 × 2 table of probabilities • Full joint probability distribution covers the complete set of random variables used to describe the world • For continuous variables it is not possible to write out the entire distribution as a table, one has to examine probability dens ...
... • E.g., P(Weather, Cavity) is a 4 × 2 table of probabilities • Full joint probability distribution covers the complete set of random variables used to describe the world • For continuous variables it is not possible to write out the entire distribution as a table, one has to examine probability dens ...
Lecture 4 1 Balls and bins games - IC
... The problems discussed before have been extensively studied in mathematics for many decades, while the problem of analyzing the maximum load has been extensively studied only very recently, because of its many applications in computer science. Indeed, the problem of finding maximum load has many app ...
... The problems discussed before have been extensively studied in mathematics for many decades, while the problem of analyzing the maximum load has been extensively studied only very recently, because of its many applications in computer science. Indeed, the problem of finding maximum load has many app ...