HW1-HW4
... (T) coming down after a fair coin is tossed are fifty-fifty. If a fair coin is tossed ten times, then intuition says that five heads are likely to turn up. Calculate the probability of getting exactly five heads (and hence exactly five tails). Solution: There are 210 possible outcomes for ten coin ...
... (T) coming down after a fair coin is tossed are fifty-fifty. If a fair coin is tossed ten times, then intuition says that five heads are likely to turn up. Calculate the probability of getting exactly five heads (and hence exactly five tails). Solution: There are 210 possible outcomes for ten coin ...
Full text PDF - IEJME-Mathematics Education
... Mathematical practices, objects and semiotic conflicts In this research we will try to explain the prospective teachers’ difficulties in computing simple, compound and conditional probabilities in the task given to them using two assumptions. Firstly we will use the idea of semiotic conflict defined ...
... Mathematical practices, objects and semiotic conflicts In this research we will try to explain the prospective teachers’ difficulties in computing simple, compound and conditional probabilities in the task given to them using two assumptions. Firstly we will use the idea of semiotic conflict defined ...
Introduction to Information Theory and Its Applications
... Definition A random variable on a probability space hΩ, B, P i is a function X : Ω → R (assigning to any elementary event ω its numeric characteristic X(ω) ∈ R we are interested in) such that for each x ∈ R we have {ω | X(ω) ≤ x} ∈ B, i.e. {ω | X(ω) ≤ x} is an event of the corresponding measurable s ...
... Definition A random variable on a probability space hΩ, B, P i is a function X : Ω → R (assigning to any elementary event ω its numeric characteristic X(ω) ∈ R we are interested in) such that for each x ∈ R we have {ω | X(ω) ≤ x} ∈ B, i.e. {ω | X(ω) ≤ x} is an event of the corresponding measurable s ...
Computing the Distribution of the Product of Two Continuous
... of the product of the coordinates of the southeast and northwest corners of the product space [i.e., (b, c) and (a, d)] when it lies entirely in the first quadrant. In order to apply the theorem to any continuous random variables X and Y , three generalizations need to be addressed. 1. Analogous the ...
... of the product of the coordinates of the southeast and northwest corners of the product space [i.e., (b, c) and (a, d)] when it lies entirely in the first quadrant. In order to apply the theorem to any continuous random variables X and Y , three generalizations need to be addressed. 1. Analogous the ...
Tilburg University Generalized Probability
... lead to the same limiting distribution for Mn as the one under H0 . Taking a substantially larger class G can improve the power of the tests and lead to tests which have similar power properties as in the one-dimensional case, but then the (asymptotic) distribution-freeness under H0 will be lost. ...
... lead to the same limiting distribution for Mn as the one under H0 . Taking a substantially larger class G can improve the power of the tests and lead to tests which have similar power properties as in the one-dimensional case, but then the (asymptotic) distribution-freeness under H0 will be lost. ...
Ch_ 5 Student Notes
... The mathematics of chance is called probability. Probability is the topic of this chapter. Here is an Activity that gives you some idea of what lies ahead. Activity: The “1 in 6 Wins” Game Let 1 through 5 represent “Please try again!” and 6 represent “You’re a winner!” 1. Roll your die seven times ...
... The mathematics of chance is called probability. Probability is the topic of this chapter. Here is an Activity that gives you some idea of what lies ahead. Activity: The “1 in 6 Wins” Game Let 1 through 5 represent “Please try again!” and 6 represent “You’re a winner!” 1. Roll your die seven times ...
A More Rational Model of Categorization Adam N. Sanborn ()
... rich structures that emerge as we learn more about our environment. Accordingly, a crucial aspect of the model is the method by which stimuli are assigned to clusters. There are two steps involved in defining any rational model of cognition: first, identifying the underlying computational problem, a ...
... rich structures that emerge as we learn more about our environment. Accordingly, a crucial aspect of the model is the method by which stimuli are assigned to clusters. There are two steps involved in defining any rational model of cognition: first, identifying the underlying computational problem, a ...
+ P(B)
... discover that there is incredible order, but also variation therein. • Probability theory seeks to describe the variation or randomness within order so that underlying order may be better understood. • Once understood, strategies can be more effectively formulated and their risks evaluated. ...
... discover that there is incredible order, but also variation therein. • Probability theory seeks to describe the variation or randomness within order so that underlying order may be better understood. • Once understood, strategies can be more effectively formulated and their risks evaluated. ...
Unit 12: Extensions and Applications
... replaces it if it’s not a white pair, so the probabilities won’t change. The events are independent, because each removed outcome is replaced. The earlier events don’t change the probabilities for later events. We could find the sample and event space for the entire trial and use the ratio. However, ...
... replaces it if it’s not a white pair, so the probabilities won’t change. The events are independent, because each removed outcome is replaced. The earlier events don’t change the probabilities for later events. We could find the sample and event space for the entire trial and use the ratio. However, ...
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.