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... The probability distribution of a random variable X lists the values and their probabilities: Requirements of a Discrete Random Variable The probabilities pi must add up to 1. Every probability pi is a number between 0 and 1. A basketball player shoots three free throws. The random variable X is th ...
... The probability distribution of a random variable X lists the values and their probabilities: Requirements of a Discrete Random Variable The probabilities pi must add up to 1. Every probability pi is a number between 0 and 1. A basketball player shoots three free throws. The random variable X is th ...
Solutions to Problem Set #7 Section 8.1 1. A fair coin is tossed 100
... table? Does it assure you that your losses will be small? Does it assure you that if n is very large you will lose? The Law of Large Numbers does NOT assure your losses will be small. In fact, it says that your losses will average .0141 per game, and hence your total losses will become arbitrarily l ...
... table? Does it assure you that your losses will be small? Does it assure you that if n is very large you will lose? The Law of Large Numbers does NOT assure your losses will be small. In fact, it says that your losses will average .0141 per game, and hence your total losses will become arbitrarily l ...
Stats random variables
... You invest 20% of your funds in Treasury bills and 80% in an “index fund” that represents all U.S. common stocks. Your rate of return over time is proportional to that of the T-bills (X) and of the index fund (Y), such that R = 0.2X + 0.8Y. Based on annual returns between 1950 and 2003: ...
... You invest 20% of your funds in Treasury bills and 80% in an “index fund” that represents all U.S. common stocks. Your rate of return over time is proportional to that of the T-bills (X) and of the index fund (Y), such that R = 0.2X + 0.8Y. Based on annual returns between 1950 and 2003: ...
Section 3.1 Functions
... (a) Use the optical reader at the checkout counter of the supermarket to convert codes to prices. Solution: For each code, the reader produces exactly one price, so this is a function. (b) The correspondence between a computer, x, and several users of the computer, y. Solution: Since for a computer ...
... (a) Use the optical reader at the checkout counter of the supermarket to convert codes to prices. Solution: For each code, the reader produces exactly one price, so this is a function. (b) The correspondence between a computer, x, and several users of the computer, y. Solution: Since for a computer ...
The Length of a Line Segment
... Recall: In section 1.5 (when we learned about Infinite Geometric series), you learned a formula to find the sum of an infinite Geometric series... and the restrictions on the formula were… ...
... Recall: In section 1.5 (when we learned about Infinite Geometric series), you learned a formula to find the sum of an infinite Geometric series... and the restrictions on the formula were… ...
Section 1.5 – The Intermediate Value Theorem.jnt
... graph of f must intersect y = N somewhere (at least once). There is a hole on the graph of function f(x) at x = -2. We say this function is NOT continuous at x = -2. ...
... graph of f must intersect y = N somewhere (at least once). There is a hole on the graph of function f(x) at x = -2. We say this function is NOT continuous at x = -2. ...
Guided Notes: Comparing and Ordering Integers, Absolute Value
... Think that negative numbers are how much you _________ and positive numbers are how much you ________. You can also think about money to help you. When comparing, you use symbols: ...
... Think that negative numbers are how much you _________ and positive numbers are how much you ________. You can also think about money to help you. When comparing, you use symbols: ...
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).