5.5.3 Convergence in Distribution
... at all points x where FX (x) is continuous. Example (Maximum of uniforms) If X1 , X2 , . . . are iid uniform(0,1) and X(n) = max1≤i≤n Xi , let us examine if X(n) converges in distribution. As n → ∞, we have for any ² > 0, P (|Xn − 1| ≥ ²) = P (X(n) ≤ 1 − ²) = P (Xi ≤ 1 − ², i = 1, . . . , n) = (1 − ...
... at all points x where FX (x) is continuous. Example (Maximum of uniforms) If X1 , X2 , . . . are iid uniform(0,1) and X(n) = max1≤i≤n Xi , let us examine if X(n) converges in distribution. As n → ∞, we have for any ² > 0, P (|Xn − 1| ≥ ²) = P (X(n) ≤ 1 − ²) = P (Xi ≤ 1 − ², i = 1, . . . , n) = (1 − ...
E2 - KFUPM AISYS
... 2) DO NOT round your answers at each step. Round answers only if necessary at your final step to 4 decimal places. 3) You are allowed to use electronic calculators and other reasonable writing accessories that help write the exam. Try to define events, formulate problem and solve. 4) Do not keep you ...
... 2) DO NOT round your answers at each step. Round answers only if necessary at your final step to 4 decimal places. 3) You are allowed to use electronic calculators and other reasonable writing accessories that help write the exam. Try to define events, formulate problem and solve. 4) Do not keep you ...
2.2 Let E and F be two events for which one knows that the
... 3.5 A ball is drawn at random from an urn containing one red and one white ball. If the white ball is drawn, it is put back into the urn. If the red ball is drawn, it is returned to the urn together with two more red balls. Then a second draw is made. What is the probability a red ball was drawn on ...
... 3.5 A ball is drawn at random from an urn containing one red and one white ball. If the white ball is drawn, it is put back into the urn. If the red ball is drawn, it is returned to the urn together with two more red balls. Then a second draw is made. What is the probability a red ball was drawn on ...
Basic Probability Statistics
... If two people are randomly selected, what is the probability both are left handed? P (both left handed) = 0:132 ' 0:016 9 ...
... If two people are randomly selected, what is the probability both are left handed? P (both left handed) = 0:132 ' 0:016 9 ...
THE LAW OF LARGE NUMBERS and Part IV N. H. BINGHAM
... his new integral, the Lebesgue integral. It turns out that this was also the mathematics of probability, with the integral as expectation. Emile BOREL (1871-1956); Borel’s Normal Number Theorem of 1909. Almost all numbers are normal to all bases simultaneously (each digit in their decimal expansions ...
... his new integral, the Lebesgue integral. It turns out that this was also the mathematics of probability, with the integral as expectation. Emile BOREL (1871-1956); Borel’s Normal Number Theorem of 1909. Almost all numbers are normal to all bases simultaneously (each digit in their decimal expansions ...
2014 Q1 Exam Review
... Explain how this demonstrates Simpson’s Paradox Why does Simpson’s Paradox occur in this situation? ...
... Explain how this demonstrates Simpson’s Paradox Why does Simpson’s Paradox occur in this situation? ...
Empirical Probability
... 14) A spinner has regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3? ...
... 14) A spinner has regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3? ...
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.