Distribution of the Sample Mean and Linear Combinations Examples Example 1 Let X1 ; X2 ; : : : ; X100 denote the actual net weights of 100 randomly selected 50-pound bags of fertilizer. a. If the expected weight of each bag is 50 pounds and the standard d

Summary Notes - Probability Theory
Basic De.nitions De.nition 1 The term probability refers to the study of randomness and uncertainty. In any situation in which one of a number of possible outcomes may occur, the theory of probability provides methods fo

More Probability Examples (Probability Reviewed)
Example 1 Each of a sample of four home mortgages is classi.ed as .xed rate (F ) or variable rate (V ). a. What are the 16 outcomes in S? Solution S = cfw_F F F F , F F F V , F F V F , F V F F , F V F V , F

Summary Notes: Poisson Distribution A random variable X is said to have a Poisson distribution if the probability mass function is: p(x; ) = for x = 0; 1; 2; : and some > 0. Note 1 The Poisson distribution is useful in modelling the number of events that

More Examples, Poisson Distribution Example 1 Suppose that the number X of tornados observed in a particular region during a one-year period has a Poisson distribution with = 8. a. Compute P (X 5), P (6 X 9), and P (10 X). Solution The desired probabiliti

Normal and Lognormal Distributions De.nitions/Notes De.nition 1 A continuous random variable X is said to have a normal distribution with parameters and (or and 2 ), where 1 < < 1 and 0 < , if the pdf of X is: 1 (x )2 , f (x; ; ) = p exp 2 2 2 for 1 < x <

Joint Probability Distributions De.nitions/Notes De.nition 1 Let X and Y be two discrete random variables de.ned on the sample space S of an experiment. The joint probability mass function p(x; y) is de.ned for each pair of numbers (x; y) by: p(x; y) = P

Joint Probability Distributions Examples Example 1 A certain market has both an express checkout line and a superexpress checkout line. Let X denote the number of customers in line at the express checkout line at a particular time of day and let Y denote

Other Continuous Distributions Notes De.nition 1 For > 0, the gamma function () is de.ned by: () = Properties of the Gamma Function 1. For any > 1, () = ( 1) ( 1). 2. For any positive integer n, (n) = (n 1)! p 1 3. ( 2 ) = De.nition 2 A continuous random

Gamma & Other Continuous Distributions Examples Example 1 Suppose the time spent by a randomly selected student who uses a terminal connected to a local time-sharing computer facility has a gamma distribution with mean 20 minutes and variance 80 minutes2

Estimation De.nitions/Notes De.nition 1 A point estimate of a parameter is a single number that can be regarded as the most plausible value of . A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample d

Estimation More Examples Example 1 Let X1 ; X2 ; :; Xn be a random sample from the Poisson distribution with mean . a. Find a point estimator for using the .rst moment with the method of moments technique. Solution The .rst moment is: E(X) = : By equating

De.nitions and other formulas - Sections 3.1 to 3.3 De.nition 1 A random variable is any rule which associates a number with each outcome in a sample space. De.nition 2 Any random variable whose only possible values are 0 and 1 is called a Bernoulli rando

More Examples (Sections 3.1 to 3.3 Reviewed) Example 1 The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables. a. T = the total number of pumps

Continuous Random Variables De.nitions/Notes De.nition 1 A random variable X is said to be continuous if its set of possible values is an entire interval of numbers that is, if for some A < B, any number x between A and B is possible. De.nition 2 Let X be

More Examples- Continuous Random Variables Example 1 Suppose the reaction temperature X (in C) in a certain chemical process has a uniform distribution with A = 5 and B = 5. a. Compute P (X < 0), P (2 < X < 2), and P (2 X 3). Solution The pdf for X is: f

Distribution of the Sample Mean and Linear Combinations De.nitions/Notes De.nition 1 A statistic is any quantity whose value can be calculated from sample data. A statistic is a random variable and will be denoted by an uppercase letter; a lowercase lette

Chapter Four Continuous Random Variables and Probability Distributions
4.1 Continuous Random Variables and Probability Density Functions For a discrete random variable X, one can assign a probability to each value that X can take (i.e., through the probab

Chapter Three Discrete Random Variables and Probability Distributions
3.1 Random Variables A random variable is any rule which associates a number with each outcome in the sample space S. Alternatively, it may be defined as a variable whose value is a num

Brief Review of Calculus
The Derivative
Definition The derivative of the function f is the function f defined by: f (x) = lim f (x + h) - f (x) , h0 h
for all x for which the limit exists. If y = f (x), we often write: dy = Dy = f (x). dx Derivative of a

Summary Notes: Binomial, Hypergeometric & Negative Binomial De.nition 1 A binomial experiment is an experiment which satis.es each of the following conditions: 1. The experiment consists of a sequence of n trials, where n is .xed in advance of the experim

More Examples: Binomial, Hypergeometric, Negative Binomial Example 1 When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X = the number of defective boards in a random sample of