LAB 3
... c) The C.L.T. is still true even if the Yi 's are from different probability distributions! All that is required for the C.L.T. to hold is that the distribution(s) have a finite mean(s) and variance(s) and that no one term in the sum dominates the sum. This is more general than definition II). 1) In ...
... c) The C.L.T. is still true even if the Yi 's are from different probability distributions! All that is required for the C.L.T. to hold is that the distribution(s) have a finite mean(s) and variance(s) and that no one term in the sum dominates the sum. This is more general than definition II). 1) In ...
Unit 7 Study Guide
... Suppose you use five different letters from the 26 letters of the alphabet to make a password. Find the number of possible five-letter passwords if letters cannot repeat. Suppose a license plate consists of five different letters. a. How many five-letter license plates are possible? b. In how many w ...
... Suppose you use five different letters from the 26 letters of the alphabet to make a password. Find the number of possible five-letter passwords if letters cannot repeat. Suppose a license plate consists of five different letters. a. How many five-letter license plates are possible? b. In how many w ...
Assignment 4
... Section and problem numbers refer to the 3rd edition of the Ghahramani textbook. 1. Section 10.1, #9. 2. Section 10.1, #15. 3. Section 10.1, #18. There is a typo in the problem statement: the sequence X1 , X2 , . . . should be an infinite sequence, not a finite one. Hint: Theorem 10.2 in the text ma ...
... Section and problem numbers refer to the 3rd edition of the Ghahramani textbook. 1. Section 10.1, #9. 2. Section 10.1, #15. 3. Section 10.1, #18. There is a typo in the problem statement: the sequence X1 , X2 , . . . should be an infinite sequence, not a finite one. Hint: Theorem 10.2 in the text ma ...
3_1
... Random Variables Random variables are customarily denoted by uppercase letters, such as X and Y, near the end of our alphabet. In contrast to our previous use of a lowercase letter, such as x, to denote a variable, we will now use lowercase letters to represent some particular value of the correspo ...
... Random Variables Random variables are customarily denoted by uppercase letters, such as X and Y, near the end of our alphabet. In contrast to our previous use of a lowercase letter, such as x, to denote a variable, we will now use lowercase letters to represent some particular value of the correspo ...
Unit 3 PowerPoint
... A fair die is rolled 10 times. Let X be the number of rolls in which we see at least one 2. What is the probability of seeing at least one 2 in any one roll of the pair of dice? What is the probability that in exactly half of the 10 rolls, we see at least one 2? ...
... A fair die is rolled 10 times. Let X be the number of rolls in which we see at least one 2. What is the probability of seeing at least one 2 in any one roll of the pair of dice? What is the probability that in exactly half of the 10 rolls, we see at least one 2? ...
MS Word - David Michael Burrow
... Last week we learned about permutations, which meant choosing a small group out of a large group in different orders. Combinations also involve choosing a small group out of a larger group. The difference is that with combinations you don’t care about the order. COMBINATIONS—it doesn’t matter ...
... Last week we learned about permutations, which meant choosing a small group out of a large group in different orders. Combinations also involve choosing a small group out of a larger group. The difference is that with combinations you don’t care about the order. COMBINATIONS—it doesn’t matter ...
Chapter 14: From Randomness to Probability
... ● the probability allows us to see general outcomes that would happen in the long run ● independent trial - outcome of one trial doesn’t influence outcome of another ● Law of Large Numbers (LLN) - long-run relative frequency gets closer and closer to true relative frequency as the number of trials i ...
... ● the probability allows us to see general outcomes that would happen in the long run ● independent trial - outcome of one trial doesn’t influence outcome of another ● Law of Large Numbers (LLN) - long-run relative frequency gets closer and closer to true relative frequency as the number of trials i ...
Infinite Sets
... positive integer; i.e., the set of rationals between 0 and 1 is countably infinite. The correspondence displayed in the array above is called an enumeration of the rationals in 0, 1. This is not the only possible enumeration of this set. To illustrate how counterintuitive infinite sets can be, we ...
... positive integer; i.e., the set of rationals between 0 and 1 is countably infinite. The correspondence displayed in the array above is called an enumeration of the rationals in 0, 1. This is not the only possible enumeration of this set. To illustrate how counterintuitive infinite sets can be, we ...
The probability of an event, expressed as P(event), is always a
... The probability of an event, expressed as P(event), is always a number between 0 and 1 (inclusive). A probability of 0 means the event is impossible and a probability of 1 means the event is certain. ...
... The probability of an event, expressed as P(event), is always a number between 0 and 1 (inclusive). A probability of 0 means the event is impossible and a probability of 1 means the event is certain. ...
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.