Notes Sept 23 and 25
... Random variables In a random process, a random variable is a function from the sample space to the real numbers. So the random variables takes a numerical value which depends on the outcome. In this course, we will only consider random variables that take nonnegative, integer values. Examples: • A d ...
... Random variables In a random process, a random variable is a function from the sample space to the real numbers. So the random variables takes a numerical value which depends on the outcome. In this course, we will only consider random variables that take nonnegative, integer values. Examples: • A d ...
Sketch
... the normal approximation for finding P(more than 1500 greet their dog first). Show Work!! How many owners would you expect to greet their dog first? ...
... the normal approximation for finding P(more than 1500 greet their dog first). Show Work!! How many owners would you expect to greet their dog first? ...
Some important discrete random variables
... Using a similar argument we can get that E[X 2 ] = λ(λ + 1), and consequently we get Var(X) = λ. Another use of the Poisson distribution arises in situations where “events” occur at certain points in time. One example of this is that an event is the occurrence of an earthquake, another possibility w ...
... Using a similar argument we can get that E[X 2 ] = λ(λ + 1), and consequently we get Var(X) = λ. Another use of the Poisson distribution arises in situations where “events” occur at certain points in time. One example of this is that an event is the occurrence of an earthquake, another possibility w ...
Statistics 510: Notes 7
... So far we have considered discrete random variables that can take on a finite or countably infinite number of values. In applications, we are often interested in random variables that can take on an uncountable continuum of values; we call these continuous random variables. Example: Consider modelin ...
... So far we have considered discrete random variables that can take on a finite or countably infinite number of values. In applications, we are often interested in random variables that can take on an uncountable continuum of values; we call these continuous random variables. Example: Consider modelin ...
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.