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22C:19 Discrete Math
... EXAMPLE 2. A fair coin is flipped three times. Let X be the random variable that assigns to an outcome the number of heads that is the outcome. What is the expected value of X? There are eight possible outcomes when a fair coin is flipped three times. These are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT ...
... EXAMPLE 2. A fair coin is flipped three times. Let X be the random variable that assigns to an outcome the number of heads that is the outcome. What is the expected value of X? There are eight possible outcomes when a fair coin is flipped three times. These are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT ...
Problem Set C
... 5. A spinner is spun 2 times. What is the probability that the sum of the two numbers is prime? ...
... 5. A spinner is spun 2 times. What is the probability that the sum of the two numbers is prime? ...
Section 8.4: Random Variables and probability distributions of
... Section 8.4: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as the mean, the median, the range and the standard deviation. When we perform an experiment, we are ...
... Section 8.4: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as the mean, the median, the range and the standard deviation. When we perform an experiment, we are ...
A random variable is a variable “whose value is a numerical
... individual number as you go “up” the x-axis. In any continuous distribution the probability of finding something requires you to find the area under a curve. So, in this case what you need to do is find the area under the red line between 0 and (1/2) on the x-axis. So, the percent of observations t ...
... individual number as you go “up” the x-axis. In any continuous distribution the probability of finding something requires you to find the area under a curve. So, in this case what you need to do is find the area under the red line between 0 and (1/2) on the x-axis. So, the percent of observations t ...
Chapter 7 problems
... Problem 5. Bo assumes that X, the height in meters of any Canadian selected by an equally likely choice among all Canadians, is a random variable with E[X] = h. Because Bo is sure that no Canadian is taller than 3 meters, he decides to use 1.5 meters as a conservative value for the standard deviatio ...
... Problem 5. Bo assumes that X, the height in meters of any Canadian selected by an equally likely choice among all Canadians, is a random variable with E[X] = h. Because Bo is sure that no Canadian is taller than 3 meters, he decides to use 1.5 meters as a conservative value for the standard deviatio ...
Discrete Probability
... EXAMPLE 2. A fair coin is flipped three times. Let X be the random variable that assigns to an outcome the number of heads that is the outcome. What is the expected value of X? There are eight possible outcomes when a fair coin is flipped three times. These are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT ...
... EXAMPLE 2. A fair coin is flipped three times. Let X be the random variable that assigns to an outcome the number of heads that is the outcome. What is the expected value of X? There are eight possible outcomes when a fair coin is flipped three times. These are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT ...
simulations, sampling distributions, probability and random variables
... Two events, A and B, are independent if the occurrence or non-occurrence of one of the events has no effect on the probability that the other event occurs. In symbols, P(A|B) = P(A) and P(B|A) = P(B) if and only if events A and B are independent. Two tests for independence: P(A | B) P(A) or P (A ...
... Two events, A and B, are independent if the occurrence or non-occurrence of one of the events has no effect on the probability that the other event occurs. In symbols, P(A|B) = P(A) and P(B|A) = P(B) if and only if events A and B are independent. Two tests for independence: P(A | B) P(A) or P (A ...
Hints on PROBABILITY probability_hints
... Two events, A and B, are independent if the occurrence or non-occurrence of one of the events has no effect on the probability that the other event occurs. In symbols, P(A|B) = P(A) and P(B|A) = P(B) if and only if events A and B are independent. Two tests for independence: P(A | B) P(A) or P (A ...
... Two events, A and B, are independent if the occurrence or non-occurrence of one of the events has no effect on the probability that the other event occurs. In symbols, P(A|B) = P(A) and P(B|A) = P(B) if and only if events A and B are independent. Two tests for independence: P(A | B) P(A) or P (A ...
Lecture 31: The law of large numbers
... Almost every real number is normal The reason is that we can look at the k’th digit of a number as the value of a random variable Xk (ω) where ω ∈ [0, 1]. These random variables are all independent and have ( the same ...
... Almost every real number is normal The reason is that we can look at the k’th digit of a number as the value of a random variable Xk (ω) where ω ∈ [0, 1]. These random variables are all independent and have ( the same ...
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).