Probability
... of many samples to see how confident we are that a sample statistic is close to the population parameter. We can compute a confidence interval around a sample mean or a proportion: – We can pick how confident we want to be – Usually choose 95%, or two standard errors – Remember, with a normal distri ...
... of many samples to see how confident we are that a sample statistic is close to the population parameter. We can compute a confidence interval around a sample mean or a proportion: – We can pick how confident we want to be – Usually choose 95%, or two standard errors – Remember, with a normal distri ...
Continuous probability
... will have various values. If we imagine that T can be any non-negative real number, then the sample space is the set of all non-negative real numbers, i.e. S = {t: t 0}. This is an uncountable set. In situations such as this, the probability that T assumes any particular value may be 0, so we are ...
... will have various values. If we imagine that T can be any non-negative real number, then the sample space is the set of all non-negative real numbers, i.e. S = {t: t 0}. This is an uncountable set. In situations such as this, the probability that T assumes any particular value may be 0, so we are ...
Lecture Note 7
... Ex. A store sells 2 different brands of DVD players. Of its DVD player sales, 60% are brand A (less expensive) and 40% are brand B. Each manufacturer offers a 1-yr warranty on parts and labor. It is known that 25% of brand A’s DVD players require warranty repair work, whereas 10% for brand B. (a) W ...
... Ex. A store sells 2 different brands of DVD players. Of its DVD player sales, 60% are brand A (less expensive) and 40% are brand B. Each manufacturer offers a 1-yr warranty on parts and labor. It is known that 25% of brand A’s DVD players require warranty repair work, whereas 10% for brand B. (a) W ...
311 review sheet. The exam covers sections 3.4, 3.5, 3.6, 3.7, 3.8
... 7. Let f (x) = cx(1 − x) for x in [0, 1] and 0 otherwise. Let X be a random variable with this density. (a) Find c (b) Find P(X > 1/2) (c) Find P(X ≤ 1/4). (d) Find E(X) (e) Find VAR(X). 8. Suppose X is uniform over the interval [10, 15]. Find VAR(X). 9. 10 random numbers are chosen uniformly from ...
... 7. Let f (x) = cx(1 − x) for x in [0, 1] and 0 otherwise. Let X be a random variable with this density. (a) Find c (b) Find P(X > 1/2) (c) Find P(X ≤ 1/4). (d) Find E(X) (e) Find VAR(X). 8. Suppose X is uniform over the interval [10, 15]. Find VAR(X). 9. 10 random numbers are chosen uniformly from ...
Handout 9 - UIUC Math
... Proposition 7 Let Z1 , Z2 , . . . be a sequence of random variables with distribution functions FZn and moment generating functions MZn , n ≥ 1; let Z be a random variable with distribution function FZ and moment generating function MZ . If MZn (t) → MZ (t) for all t, then FZn (t) → FZ (t) for all t ...
... Proposition 7 Let Z1 , Z2 , . . . be a sequence of random variables with distribution functions FZn and moment generating functions MZn , n ≥ 1; let Z be a random variable with distribution function FZ and moment generating function MZ . If MZn (t) → MZ (t) for all t, then FZn (t) → FZ (t) for all t ...
General Probability, I: Rules of probability
... consistent with this rule (assuming areas are normalized so that the entire sample space S has area 1), and you can derive each of these rules simply by calculating areas in Venn diagrams. Moreover, the area trick makes it very easy to find probabilities in other, more complicated, situations. for e ...
... consistent with this rule (assuming areas are normalized so that the entire sample space S has area 1), and you can derive each of these rules simply by calculating areas in Venn diagrams. Moreover, the area trick makes it very easy to find probabilities in other, more complicated, situations. for e ...
4 Conditional Probability - Notes
... contestants were given a choice of 3 curtains. They chose one and the host, Monty Hall, would show them a ZONK! that was behind one of the doors that they did not choose. They were then given the opportunity to switch doors. What should they do and why? Monty Hall Problem ...
... contestants were given a choice of 3 curtains. They chose one and the host, Monty Hall, would show them a ZONK! that was behind one of the doors that they did not choose. They were then given the opportunity to switch doors. What should they do and why? Monty Hall Problem ...
Hints on PROBABILITY probability_hints
... As sample size gets larger, the shape of the sampling distribution gets more normal. Large is defined as both n and n(1 ) 10 . The properties are the same as for sample means because a proportion is a special kind of mean. ...
... As sample size gets larger, the shape of the sampling distribution gets more normal. Large is defined as both n and n(1 ) 10 . The properties are the same as for sample means because a proportion is a special kind of mean. ...
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.