Discrete Random Variables
... • Definition: Mathematically, a random variable (r.v.) on a sample space S is a function1 from S to the real numbers. More informally, a random variable is a numerical quantity that is “random”, in the sense that its value depends on the outcome of a random experiment. • Notation: One commonly uses ...
... • Definition: Mathematically, a random variable (r.v.) on a sample space S is a function1 from S to the real numbers. More informally, a random variable is a numerical quantity that is “random”, in the sense that its value depends on the outcome of a random experiment. • Notation: One commonly uses ...
PHY 4105: Quantum Information Theory Lecture 2
... The central limit theorem tells you a bit more than what the weak law of large numbers does. It tells you not only the mean and the variance of the sample mean but it tells that the sample mean is a random variable distributed according to the Normal distribution. Let us push things a bit forward an ...
... The central limit theorem tells you a bit more than what the weak law of large numbers does. It tells you not only the mean and the variance of the sample mean but it tells that the sample mean is a random variable distributed according to the Normal distribution. Let us push things a bit forward an ...
Glossary Term Definition equiangular triangle A triangle with three
... A specific outcome or type of outcome. Any values of a variable that result in a denominator of 0 must be excluded from the domain of that variable. The sum of the products of each digit and its place value of a number. Example: 867 = 800 + 60 + 7 The ratio of the number of positive outcomes to the ...
... A specific outcome or type of outcome. Any values of a variable that result in a denominator of 0 must be excluded from the domain of that variable. The sum of the products of each digit and its place value of a number. Example: 867 = 800 + 60 + 7 The ratio of the number of positive outcomes to the ...
Math 151 Midterm 2 Solutions
... An airline company sells 200 tickets for a plane with 198 seats, knowing that the probability a passenger will not show up for the flight is 0.01. Use the Poisson approximation to compute the probability they will have enough seats for all passengers who show up. Solution. In the language of Poisson ...
... An airline company sells 200 tickets for a plane with 198 seats, knowing that the probability a passenger will not show up for the flight is 0.01. Use the Poisson approximation to compute the probability they will have enough seats for all passengers who show up. Solution. In the language of Poisson ...
9. DISCRETE PROBABILITY DISTRIBUTIONS
... a sequence of zeros and ones, such as 1 1 0 1 0. Each of the five digits in this sequence represents the outcome of the random experiment of tossing a die once, where 1 denotes Heads and 0 denotes Tails. We have five repetitions of the experiment. • A random variable is discrete if it can assume onl ...
... a sequence of zeros and ones, such as 1 1 0 1 0. Each of the five digits in this sequence represents the outcome of the random experiment of tossing a die once, where 1 denotes Heads and 0 denotes Tails. We have five repetitions of the experiment. • A random variable is discrete if it can assume onl ...
4. DISCRETE PROBABILITY DISTRIBUTIONS
... What does “countably infinite” mean? We won’t try to define this precisely, but an important example (the only one we will consider) of a countably infinite set is the nonnegative integers, ...
... What does “countably infinite” mean? We won’t try to define this precisely, but an important example (the only one we will consider) of a countably infinite set is the nonnegative integers, ...
AP Stats CH7 Combining Random Variables
... Question: If we profit $2000 for every radio, what will be our expected total profit? ...
... Question: If we profit $2000 for every radio, what will be our expected total profit? ...
doc - Berkeley Statistics
... exclusive, and their probabilities must add up to 1. This kind of X is called a “discrete random variable”, and has a discrete probability distribution. Here are the examples for such kind of X’s: X can be the number of calls you are expecting to receive from 2:00pm-4:00pm today. Or X is defined as ...
... exclusive, and their probabilities must add up to 1. This kind of X is called a “discrete random variable”, and has a discrete probability distribution. Here are the examples for such kind of X’s: X can be the number of calls you are expecting to receive from 2:00pm-4:00pm today. Or X is defined as ...
Working with Probability ~ 2
... • Continuous r.v. (height, mass, time, …) – variable takes only continuous values – variable defined over a range of values a ! x ! b so an infinite number of possible values for a ≠ b – the pdf is now represented by a function f(x) and probabilities are determined by the area under the curve given ...
... • Continuous r.v. (height, mass, time, …) – variable takes only continuous values – variable defined over a range of values a ! x ! b so an infinite number of possible values for a ≠ b – the pdf is now represented by a function f(x) and probabilities are determined by the area under the curve given ...
expected value - Ursinus College Student, Faculty and Staff Web
... Two random variables X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the occurrence of any event involving Y alone. An example is the rolling of two dice. Dealings in a game of blackjack is an example of dependence of events. We may ass ...
... Two random variables X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the occurrence of any event involving Y alone. An example is the rolling of two dice. Dealings in a game of blackjack is an example of dependence of events. We may ass ...
Discrete and Continuous Random Variables
... Discrete and Continuous Random Variables Discrete : A random variable is called a discrete random variable if its set of possible outcomes is countable. This usually occurs for any random variable which is a count of occurrences or of items, for example, the number of large diameter piles selected i ...
... Discrete and Continuous Random Variables Discrete : A random variable is called a discrete random variable if its set of possible outcomes is countable. This usually occurs for any random variable which is a count of occurrences or of items, for example, the number of large diameter piles selected i ...
PPT
... •The mean m of a probability distribution P(x) over a finite set of numbers x is defined ...
... •The mean m of a probability distribution P(x) over a finite set of numbers x is defined ...
3.6 Indicator Random Variables, and Their Means and Variances
... Definition 5 A random variable that has the value 1 or 0, according to whether a specified event occurs or not is called an indicator random variable for that event. You’ll often see later in this book that the notion of an indicator random variable is a very handy device in certain derivations. But ...
... Definition 5 A random variable that has the value 1 or 0, according to whether a specified event occurs or not is called an indicator random variable for that event. You’ll often see later in this book that the notion of an indicator random variable is a very handy device in certain derivations. But ...
Expected value a weighted average of all possible values where the
... Expected Value • If a random variable x can have any of the values x1, x2 , x3 ,… • the corresponding probabilities of these values occurring are P(x1), P(x2), P(x3), … • then the expected value of x is given by ...
... Expected Value • If a random variable x can have any of the values x1, x2 , x3 ,… • the corresponding probabilities of these values occurring are P(x1), P(x2), P(x3), … • then the expected value of x is given by ...
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).