Random variables - Penn Math
... Binomial trials Problem: Let X be the number of heads that occur when a coin is flipped 6 times. a) Compute the probability distribution of X . b) Compute the probability distribution of (X − 3)2 . Note that two different random variables can have the same probability distribution. For example in t ...
... Binomial trials Problem: Let X be the number of heads that occur when a coin is flipped 6 times. a) Compute the probability distribution of X . b) Compute the probability distribution of (X − 3)2 . Note that two different random variables can have the same probability distribution. For example in t ...
Maths Workshops - Probability, Sigma Notation and
... To find the number of different arrangements: 1. Select a first choice from 3 possible choices. 2. Take a second choice; there are 2 choices remaining. 3. Finally, there is 1 choice for the last selection. Thus, there are 3 × 2 × 1 = 6 different ordered arrangements of the toppings. All of these wer ...
... To find the number of different arrangements: 1. Select a first choice from 3 possible choices. 2. Take a second choice; there are 2 choices remaining. 3. Finally, there is 1 choice for the last selection. Thus, there are 3 × 2 × 1 = 6 different ordered arrangements of the toppings. All of these wer ...
The Addition Rules Mutually Exclusive Events Many problems in
... at random from a deck, what is the probability that the card is a king or a queen? In this case, there are two situations to consider. They are: The card selected is a king The card selected is a queen Now consider another example. When a card is selected from a deck, find the probability that the c ...
... at random from a deck, what is the probability that the card is a king or a queen? In this case, there are two situations to consider. They are: The card selected is a king The card selected is a queen Now consider another example. When a card is selected from a deck, find the probability that the c ...
non-uniform random variate generation
... Walker (1974, 1977) showed that one can construct in time O(N ) a table (q i , ri ), 1 ≤ i ≤ N , such that the following method works: pick a uniform integer Z from 1, 2, . . . , N . Generate a uniform [0, 1] random variate U . If U ≤ qZ , then return Z, else return rZ . The values ri are called the ...
... Walker (1974, 1977) showed that one can construct in time O(N ) a table (q i , ri ), 1 ≤ i ≤ N , such that the following method works: pick a uniform integer Z from 1, 2, . . . , N . Generate a uniform [0, 1] random variate U . If U ≤ qZ , then return Z, else return rZ . The values ri are called the ...
Random Number Generators: Metrics and Tests for Uniformity and
... In our revision, every test described here is applied to may sub-sequences from the random number generator (typically over 300), and the length of every sub-sequence is over 1,000 numbers. From all of these computations we form a random distribution which should be the same as the one described by ...
... In our revision, every test described here is applied to may sub-sequences from the random number generator (typically over 300), and the length of every sub-sequence is over 1,000 numbers. From all of these computations we form a random distribution which should be the same as the one described by ...
An Operational Characterization of the Notion of Probability by
... properties that make it clearly nonrandom. (For the development of the theory of collectives from the point of view of the definition of randomness, see Downey and Hirschfeldt [6].) In 1966, Martin-Löf [10] introduced the definition of random sequences, which is called Martin-Löf randomness nowada ...
... properties that make it clearly nonrandom. (For the development of the theory of collectives from the point of view of the definition of randomness, see Downey and Hirschfeldt [6].) In 1966, Martin-Löf [10] introduced the definition of random sequences, which is called Martin-Löf randomness nowada ...
6.5
... (e) Probability of AB is the addition of all the simple probabilities that are in BOTH A and B…in this case, there are NONE, so the probability is ZERO (f) Probability of AC is the addition of all the simple probabilities that are in BOTH A and C…in this case, there is only one P(f), so the probab ...
... (e) Probability of AB is the addition of all the simple probabilities that are in BOTH A and B…in this case, there are NONE, so the probability is ZERO (f) Probability of AC is the addition of all the simple probabilities that are in BOTH A and C…in this case, there is only one P(f), so the probab ...
Statistics
Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or societal problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as ""all persons living in a country"" or ""every atom composing a crystal"". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.Two main statistical methodologies are used in data analysis: descriptive statistics, which summarizes data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draws conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and a synthetic data drawn from idealized model. An hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is falsely rejected giving a ""false positive"") and Type II errors (null hypothesis fails to be rejected and an actual difference between populations is missed giving a ""false negative""). Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other important types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data and/or censoring may result in biased estimates and specific techniques have been developed to address these problems.Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more heavily from calculus and probability theory. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data.