Introduction to Statistics
... A batch of pregnancy test kit contains 50 kits of which 10% are known to be defective. If 3 test kits are randomly chosen with replacement from the batch, what is the probability that: (i) all will be defective; (ii) none will be defective; (iii) at least one will be defective; (iv) exactly one will ...
... A batch of pregnancy test kit contains 50 kits of which 10% are known to be defective. If 3 test kits are randomly chosen with replacement from the batch, what is the probability that: (i) all will be defective; (ii) none will be defective; (iii) at least one will be defective; (iv) exactly one will ...
Lecture8_SP16_statistical_decisions
... Gardner-Medwin : criterion is not the probability of guilt, but rather the probability of the evidence, given that the defendant is innocent (akin to a frequentist p-value). If the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. A: T ...
... Gardner-Medwin : criterion is not the probability of guilt, but rather the probability of the evidence, given that the defendant is innocent (akin to a frequentist p-value). If the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. A: T ...
Markov and Chebyshev`s Inequalities
... Recall: E(X ) = 200 ∗ (1/10) = 20. By Chernoff bounds, P(X ≥ 120) = P(X ≥ 6E(X )) ≤ 2−6E(X ) = 2−(6·20) = 2−120 . Note: By using Markov’s inequality, we were only able to determine that P(X ≥ 120) ≤ (1/6). But by using Chernoff bounds, which are specifically geared for large deviation bounds for bin ...
... Recall: E(X ) = 200 ∗ (1/10) = 20. By Chernoff bounds, P(X ≥ 120) = P(X ≥ 6E(X )) ≤ 2−6E(X ) = 2−(6·20) = 2−120 . Note: By using Markov’s inequality, we were only able to determine that P(X ≥ 120) ≤ (1/6). But by using Chernoff bounds, which are specifically geared for large deviation bounds for bin ...
Probability, Justice, and the Risk of Wrongful
... know or assume about your friend, and many other factors. In any case, it is not the same as the p-value, and indeed it could be very different. Of even greater concern is multiple testing, or what I call the Out Of How Many principle. For example, in my book6 , I tell the true story of running into ...
... know or assume about your friend, and many other factors. In any case, it is not the same as the p-value, and indeed it could be very different. Of even greater concern is multiple testing, or what I call the Out Of How Many principle. For example, in my book6 , I tell the true story of running into ...
notebook05
... rth event of a Poisson process whose rate is λ events per unit time. Then X can be thought of as the sum of r independent exponential random variables representing (1) the time to the first event, (2) the time from the first to the second event, (3) the time from the second to the third event, and s ...
... rth event of a Poisson process whose rate is λ events per unit time. Then X can be thought of as the sum of r independent exponential random variables representing (1) the time to the first event, (2) the time from the first to the second event, (3) the time from the second to the third event, and s ...
Presentation Thursday March 28
... We roll a die and let X denote the number that appears and flip a fair coin and let Y indicate Heads or Tails. ...
... We roll a die and let X denote the number that appears and flip a fair coin and let Y indicate Heads or Tails. ...
LAWS OF LARGE NUMBERS FOR PRODUCT OF RANDOM
... In the proof martingales theory, Lenglart’s inequality and properties of product integral are used (see [1]). Martingales theory is also used to prove the following result. Theorem 5 The process MSn = {MSn (t) ; t > 0}, defined by ...
... In the proof martingales theory, Lenglart’s inequality and properties of product integral are used (see [1]). Martingales theory is also used to prove the following result. Theorem 5 The process MSn = {MSn (t) ; t > 0}, defined by ...
Notes Binomial
... Removing one switch changes the proportion of bad switches remaining in the shipment. So the state of the second switch chosen is not independent of the first. But removing one switch from a shipment of 10,000 changes the makeup of the remaining 9999 switches very little. In practice, the distributi ...
... Removing one switch changes the proportion of bad switches remaining in the shipment. So the state of the second switch chosen is not independent of the first. But removing one switch from a shipment of 10,000 changes the makeup of the remaining 9999 switches very little. In practice, the distributi ...
![Q1 [20 points] Q2 [20 points]](http://s1.studyres.com/store/data/012801942_1-7e7115778c021991ef6f06b5ae7330d9-300x300.png)
Q1 [20 points] Q2 [20 points]
... true for every G1 ∈ G. Therefore, < is reflexive. b) No, < is not symmetric. A counter example. Let G = [R, +]. Both G1 = [Z, +] and G2 = [{0}, +] are subgroups of G and G1 < G2 . But G2 < G1 is not true since Z 6⊆ {0} c) Yes, < is antisymmetric. Let G1 = [A1 , ∗1 ], G2 = [A2 , ∗2 ] ∈ G with G1 < G2 ...
... true for every G1 ∈ G. Therefore, < is reflexive. b) No, < is not symmetric. A counter example. Let G = [R, +]. Both G1 = [Z, +] and G2 = [{0}, +] are subgroups of G and G1 < G2 . But G2 < G1 is not true since Z 6⊆ {0} c) Yes, < is antisymmetric. Let G1 = [A1 , ∗1 ], G2 = [A2 , ∗2 ] ∈ G with G1 < G2 ...
Representing and Finding Sequence Features using Frequency
... Goal: Describe a sequence feature (or motif) more quantitatively than possible using consensus sequences Need to describe how often particular bases are found in particular positions in a sequence feature ...
... Goal: Describe a sequence feature (or motif) more quantitatively than possible using consensus sequences Need to describe how often particular bases are found in particular positions in a sequence feature ...
Infinite Markov chains, continuous time Markov chains
... Otherwise the state is defined to be recurrent. It is defined to be positive recurrent if E[Ti ] < ∞ and null-recurrent if E[Ti ] = ∞. Thus, unlike the finite state case, the state is transient if there is a positive probability of no return, as opposed to existence of a state from which the return to ...
... Otherwise the state is defined to be recurrent. It is defined to be positive recurrent if E[Ti ] < ∞ and null-recurrent if E[Ti ] = ∞. Thus, unlike the finite state case, the state is transient if there is a positive probability of no return, as opposed to existence of a state from which the return to ...
Generative Techniques: Bayes Rule and the Axioms of Probability
... Conditional probability is the probability of an event given that another event has occurred. Conditional probability measures "correlation" or "association". Conditional probability does not measure causality. Consider two independent classes of observation A and B Let P(A) be the probability that ...
... Conditional probability is the probability of an event given that another event has occurred. Conditional probability measures "correlation" or "association". Conditional probability does not measure causality. Consider two independent classes of observation A and B Let P(A) be the probability that ...
6 - Rice University
... was no symbol for zero. Let us see for ourselves. Let us represent the numbers 1 and 60. ...
... was no symbol for zero. Let us see for ourselves. Let us represent the numbers 1 and 60. ...
Sec 28-29
... are independent. This is equivalent to saying that P ( X i ∈ A i i ≤ k) = ki=1 P( X i ∈ A i ) for any A i ∈ B (R). Events A 1 , . . . , A k are said to be independent if 1 A 1 , . . . , 1 A k are independent. This is equivalent to saying that P( A j 1 ∩ . . . ∩ A j # ) = P( A j 1 ) . . . P( A j # ) ...
... are independent. This is equivalent to saying that P ( X i ∈ A i i ≤ k) = ki=1 P( X i ∈ A i ) for any A i ∈ B (R). Events A 1 , . . . , A k are said to be independent if 1 A 1 , . . . , 1 A k are independent. This is equivalent to saying that P( A j 1 ∩ . . . ∩ A j # ) = P( A j 1 ) . . . P( A j # ) ...
possible numbers total possible numbers even . . . . 2 1 6 3 =
... 3. The probability of rolling a prime number ...
... 3. The probability of rolling a prime number ...
Gambler`s Ruin - Books in the Mathematical Sciences
... this. We have i dollars out of a total of n. We flip a coin which has probability p of landing heads, and probability q = 1 ! p of landing tails. If the coin lands heads we gain another dollar, otherwise we lose a dollar. The game continues until we have all n dollars or we are broke. The game is re ...
... this. We have i dollars out of a total of n. We flip a coin which has probability p of landing heads, and probability q = 1 ! p of landing tails. If the coin lands heads we gain another dollar, otherwise we lose a dollar. The game continues until we have all n dollars or we are broke. The game is re ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.