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... Typically, X, Y, Z etc denote random variables; x, y, z, etc denote values attained by random variables. Example: Rolling a pair of dice. Let X be the random variable corresponding to the sum of the dice on a roll. If we think of the sample points as a pair (i, j), where i = value rolled by the ...
... Typically, X, Y, Z etc denote random variables; x, y, z, etc denote values attained by random variables. Example: Rolling a pair of dice. Let X be the random variable corresponding to the sum of the dice on a roll. If we think of the sample points as a pair (i, j), where i = value rolled by the ...
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... Using past data on TV sales (below left), a tabular representation of the probability distribution for TV sales (below right) was developed. ...
... Using past data on TV sales (below left), a tabular representation of the probability distribution for TV sales (below right) was developed. ...
Section 8.2 Markov and Chebyshev Inequalities and the Weak Law
... which is the general result. EXAMPLE: An astronomer is measuring the distance to a star. Because of different errors, each measurement will not be precisely correct, but merely an estimate. He will therefore make a series of measurements and use the average as his estimate of the distance. He beli ...
... which is the general result. EXAMPLE: An astronomer is measuring the distance to a star. Because of different errors, each measurement will not be precisely correct, but merely an estimate. He will therefore make a series of measurements and use the average as his estimate of the distance. He beli ...
Expected Value and Markov Chains
... We reviewed common methods used to find the expected number of steps for a random process to reach an absorbing state in a Markov chain. A good grasp of various methods can help us discern the situation and apply the most effective method to solve the problem at hand. However, we are not limited to ...
... We reviewed common methods used to find the expected number of steps for a random process to reach an absorbing state in a Markov chain. A good grasp of various methods can help us discern the situation and apply the most effective method to solve the problem at hand. However, we are not limited to ...
Events A1,...An are said to be mutually independent if for all subsets
... numbers with heights 5’11” and 6’1”. The mean or average of this distribution is 6’, as can be determined by summing the heights of all the people and dividing by the number of people, or equivalently by summing over distinct heights weighted by the fractional number of people with that height. Supp ...
... numbers with heights 5’11” and 6’1”. The mean or average of this distribution is 6’, as can be determined by summing the heights of all the people and dividing by the number of people, or equivalently by summing over distinct heights weighted by the fractional number of people with that height. Supp ...
STATISTICAL LABORATORY, May 14th, 2010 EXPECTATIONS
... becomes more and more concentrated in a (right) neighbourhood of zero. The density plot is obtained easily through R. ...
... becomes more and more concentrated in a (right) neighbourhood of zero. The density plot is obtained easily through R. ...
Absolute Value Functions and Graphs An
... 1. Graphing an absolute value function can be accomplished using the tried and true method by selecting x or y values and solving the equation for the remaining variable and graph the x-y coordinates. 2. Absolute value functions can also be graphed on a graphing calculator: 1. Y= 2. MATH → NUM
... 1. Graphing an absolute value function can be accomplished using the tried and true method by selecting x or y values and solving the equation for the remaining variable and graph the x-y coordinates. 2. Absolute value functions can also be graphed on a graphing calculator: 1. Y= 2. MATH → NUM
Chapter 10: Math Notes
... Evaluate an Expression - The word evaluate indicates that the value of an expression should be calculated when a variable is replaced by a numerical value. For example, when you evaluate the expression x2 + 4x – 3 for x = 5, the result is: ...
... Evaluate an Expression - The word evaluate indicates that the value of an expression should be calculated when a variable is replaced by a numerical value. For example, when you evaluate the expression x2 + 4x – 3 for x = 5, the result is: ...
Expected value
In probability theory, the expected value of a random variable is intuitively the long-run average value of repetitions of the experiment it represents. For example, the expected value of a dice roll is 3.5 because, roughly speaking, the average of an extremely large number of dice rolls is practically always nearly equal to 3.5. Less roughly, the law of large numbers guarantees that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions goes to infinity. The expected value is also known as the expectation, mathematical expectation, EV, mean, or first moment.More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. The same works for continuous random variables, except the sum is replaced by an integral and the probabilities by probability densities. The formal definition subsumes both of these and also works for distributions which are neither discrete nor continuous: the expected value of a random variable is the integral of the random variable with respect to its probability measure.The expected value does not exist for random variables having some distributions with large ""tails"", such as the Cauchy distribution. For random variables such as these, the long-tails of the distribution prevent the sum/integral from converging.The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations: it is the expected value of the squared deviation of the variable's value from the variable's expected value.The expected value plays important roles in a variety of contexts. In regression analysis, one desires a formula in terms of observed data that will give a ""good"" estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, so the estimate it gives is itself a random variable. A formula is typically considered good in this context if it is an unbiased estimator—that is, if the expected value of the estimate (the average value it would give over an arbitrarily large number of separate samples) can be shown to equal the true value of the desired parameter.In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information. For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for risk averse agents it involves maximizing the expected value of some objective function such as a von Neumann-Morgenstern utility function. One example of using expected value in reaching optimal decisions is the Gordon-Loeb Model of information security investment. According to the model, one can conclude that the amount a firm spends to protect information should generally be only a small fraction of the expected loss (i.e., the expected value of the loss resulting from a cyber/information security breach).