Chapter 2 Introduction to Discrete Random Variables
... • It turns out that we can extract the marginal probability mass function pX (xi ) and pY (yj ) from the joint pmf pXY (xi , yj ) using the formulas X pX (xi ) = pXY (xi , yj ) ...
... • It turns out that we can extract the marginal probability mass function pX (xi ) and pY (yj ) from the joint pmf pXY (xi , yj ) using the formulas X pX (xi ) = pXY (xi , yj ) ...
Random Variable
... 1. Suppose that you are invited to play a game with the following rules: One of the numbers 2, … , 12 is chosen at random by throwing a pair of dice and adding the numbers shown. You win 9 dollars in case 2, 3, 11, or 12 comes out, or lose 10 dollars if the outcome is 7. Otherwise, you do not win or ...
... 1. Suppose that you are invited to play a game with the following rules: One of the numbers 2, … , 12 is chosen at random by throwing a pair of dice and adding the numbers shown. You win 9 dollars in case 2, 3, 11, or 12 comes out, or lose 10 dollars if the outcome is 7. Otherwise, you do not win or ...
Lecture 11: Random Variables
... The study of random variables is motivated by the fact that in many scenarios, one might not be interested in the precise elementary outcome of a random experiment, but rather in some numerical function of the outcome. For example, in an experiment involving ten coin tosses, the experimenter may onl ...
... The study of random variables is motivated by the fact that in many scenarios, one might not be interested in the precise elementary outcome of a random experiment, but rather in some numerical function of the outcome. For example, in an experiment involving ten coin tosses, the experimenter may onl ...
Discrete Random Variables
... variable X takes the value x. Often we will make this even shorter and simply write p(x). Example: For a six-sided die with the values {1, 2, 3, 4, 5, 6} on it’s faces, let X be the value that appears on the face when the die is rolled. Then p(2) or P(X=2) represents the probability that X will be 2 ...
... variable X takes the value x. Often we will make this even shorter and simply write p(x). Example: For a six-sided die with the values {1, 2, 3, 4, 5, 6} on it’s faces, let X be the value that appears on the face when the die is rolled. Then p(2) or P(X=2) represents the probability that X will be 2 ...
Mathematics for Business Decisions, Part II
... had come from a sample of 100 administrators, compute the 95% confidence interval for X . (ii) With the larger sample size, could the administrator claim, at the 95% level, that his salary is below the national average? Show all work.. Solution: 2. In a previous exercise, you found a number z1 su ...
... had come from a sample of 100 administrators, compute the 95% confidence interval for X . (ii) With the larger sample size, could the administrator claim, at the 95% level, that his salary is below the national average? Show all work.. Solution: 2. In a previous exercise, you found a number z1 su ...
Independent random variables
... Proposition 2. Let X and Y be continuous random variables. Let c be a real number and z = g(x) and z = h(x, y) be real valued functions. Then ...
... Proposition 2. Let X and Y be continuous random variables. Let c be a real number and z = g(x) and z = h(x, y) be real valued functions. Then ...
Solutions
... and V ar(−X) = 45.6192 (recall V ar(aX) = a2 V ar(X)). Thus we have that D is approximately normal with µD = 52 − 70.4 = −18.4 and σD = 38.48 + 35.6192. ...
... and V ar(−X) = 45.6192 (recall V ar(aX) = a2 V ar(X)). Thus we have that D is approximately normal with µD = 52 − 70.4 = −18.4 and σD = 38.48 + 35.6192. ...
Randomness
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or ""trials"") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators.Random selection is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.