Chapter 5
Properties of a Valid Probability Distribution for a Discrete Random Variable (pg.198)
*KEY*
Poisson Probability Distribution (pg
Has many practical applications and is often associated with *rare* events.
-Geometric Experiment (pg.
The number of successes is fixed at 1, and the number of trials varies.
-Mean (Expected Value) (pg.203)
1.) The capital E stands for *expected value* and is a function. The function E takes a random variable as an input and returns the expected value. 2.) m is the mean, or expected value, of a random variable (which may model a population). 3.) The mean is easy to compute. Multiply each value of the random variable by its corresponding probability, and add the products. 4.) The mean of a random variable is a *weighted average* and is only what happens on average. The mean may not be any of possible values of random variable.
Properties of a Geometric Experiment (pg.
1.) The experiment consists of identical trials. 2.) Each trial can result in only one of two possible outcomes: a success (S) or a failure (F). 3.) The trials are independent. 4.) The probability of a success, p, is constant from trial to trial. *The experiment ends when the first success is obtained.*
Properties of a Binomial Experiment (pg.212)
1.) The experiment consists of n identical trials. 2.) Each trial can result in only one of two possible (mutually exclusive) outcomes. One outcome is usually designated a success (S) and the other a failure (F). 3.) The outcomes of the trials are independent. 4.) The probability of a success, p, is constant from trial to trial. -A *trial* is a small part of the larger experiment. A trial results in a single occurrence of either a success or a failure. -A *success* does not have to be a *good* thing. -Trials are *independent* if whatever happens on one trial has no effect on any other trial. Ex. Any one voter response has no effect on any other voter response. -The probability of a success on every trial is exactly the same. Ex. The probability of the tossed (fair) coin landing with head face up is always 1/2.
Properties of a Hypergeometric Experiment (pg.
1.) The population consists of N objects, of which M are success and N-M are failures. 2.) A sample of *n* objects is select *without* replacement. 3.) Each sample of size *n* is equally likely. -The hypergeometric probability distribution is completely determined by n, N and M. -It is impossible to obtain less than 1 success. Also, the greatest number of successes possible is 5.
Properties of a Poisson Experiment (pg.
1.) The probability that a single event occurs in a given interval (of time, length, volume, etc.) is the same for all intervals. 2.) The number of events that occur in any interval is indecent of the number that occur in any other interval. *These properties are often referred to as a *Poisson process* and and can be difficult to verify.* -The Poisson distribution is completely determined by the mean, denoted by the break latter lambda. Because the Poisson distribution is often used to count rare events, the mean number of events per interval is usually small. The probability distribution is given below. -The Poisson distribution is completely characterized by only one parameter, lambda. The mean and the variance are both equal to the same value, lambda.
-Continuous (pg.190)
A random variable is continuous if the set of all possible values is an interval of numbers. -Continuous random variables are usually associated with *measuring.* -An interval of possible values means continuous. (pg.190): The interval of possible values for a continuous random variable can be *any* interval, of any length, open or closed. The exact interval may not be known, only that there is *some* interval os possible values. In practice, no measurement device is precise enough to return *any* number in some interval. In *theory,* a continuous random variable may assume any value in some interval (but not in reality). time price
-Discrete (pg.190)
A random variable is discrete if the set of all possible values is finite, or countably infinite. -Discrete random variables are usually associated with *counting.* -Finite or countably infinite means discrete. (pg.190): Countably infinite means there are infinitely many possible values, but they are countable. You may not ever be able to finish counting all of the possible values, but there exists a *method* for actually counting them.
Hypergeometric Probability Distribution (pg
Arises form an experiment in which there is sampling without replacement from a finite population. Each element in the population is labeled a success or failure.
5.5 OTHER DISCRETE DISTRIBUTIONS (PG.224-238):
In a binomial experiment, n (the number of trials) is fixed and the number of success varies. on the other hand In a geometric experiment, the number of success is fixed at 1, and the number of trials varies.
-Poisson Random Variable (pg.
Is a count of the number of occurrences of a certain event in a given unit of time, space, volume, distance, etc. ~A count of the number of times the specific event occurs during a given interval. Ex. The number of arrivals to a hospital Emergency Room in a certain 30-minute period, the number of asteroids that pass through Earth's orbit during a given year, or the number of bacteria in milliliter of drinking water.
-Hypergeometric Random Variable (pg.
Is a count of the number of successes in the random sample of size *n*. Ex. Consider a shipment of 12 automobile tires, of which two are defective, and a random sample of four tires. A hypergeometric random variable may be defined as a count of the number of *good* tires selected.
-Random Variable (pg.188)
Is a function that assigns a unique numerical value to each outcome in a sample space. ~A random variable maps elements of a sample space to the real numbers. Note that several outcomes may be assigned to the same number, but each outcome is assigned to only one number. 1.) Such functions are called random variables because their values cannot be predicted with certainty before the experiment is performed. 2.) Capital letters, such s X and Y, are used to represent random variables.
-Probability Distribution for a Discrete Random Variable X (pg.193)
Is a method for specifying *all* of the possible values of X and the probability associated with each value. 1.) May be presented in the form of an itemized listing, a table, a graph, or a function.
-Geometric Random Variable (pg.
Is the number of trials necessary to realize the first success.
5.3 MEAN, VARIANCE, AND STANDARD DEVIATION FOR A DISCRETE RANDOM VARIABLE (pg.202-211):
Just as there are descriptive measures of a sample, there are corresponding descriptive measures of a population. As we said in Chapter 3, these population *parameters* describe the center and variability of the *entire* population.
-Binomial Random Variable (pg.212)
Maps each outcome in a binomial experiment to a real number, and is defined to be the *number of successes* in *n* trials. The probability of an outcome depends on the *number of successes* (and failures), *not* on the order in which they appear. (pg.213)
5.4 THE BINOMIAL DISTRIBUTION (pg.211-224):
Properties of a Binomial Experiment (pg.212): 1.) The experiment consists of n identical trials. 2.) Each trial can result in only one of two possible (mutually exclusive) outcomes. One outcomes is usually designated a success (S) and the other a failure (F). 3.) The outcomes of the trials are independent. 4.) The probability of a success, p, is constant form trial to trial.
-Cumulative Probabilities (pg.215)
The probability that X takes on a value *less than or equal to x.* Accumulate all the probability associated with values up to ansi including x. *Every* probability question about a binomial random variable can be answered using *cumulative* probability. There may also be other, faster methods, but cumulative probability always works.
-Variance and Standard Deviation of a Random Variable (pg.204)
measure the spread of the distribution Variance is computed using the expected value function, and the standard deviation together with the mean can be used to determine the most likely values of the random variable. *Example 5.11 (pg.205-207) GREAT explanation!*
The Probability of Obtaining x Successes is (pg.213)
p(X=x) = (number of outcomes with x successes)p^x(1-p)^n-x