Statistics Chapter 5 (Bentley)

Three Characteristics of a Discrete Uniform Distribution

1. The distribution has a finite number of specified values 2. Each value is equally likely 3. The distribution is symmetric

Two Key Properties of Discrete Probability Distributions

1. The probability of each value x is a value between 0 and 1, or equivalently, 0<=P(X=x)<=1 2. The sum of the probabilities equals 1. In other words, Σ P(X=x1)=1 where the sum extends over all values x of X.

Risk Loving Consumer

A consumer that may be willing to take a risk even if the expected gain is negative.

Random Variable

A function that assigns numerical values to the outcomes of an experiment.

Expected Value (also referred to as Population Mean)

A weighted average of all possible values of X For a discrete random variable X with values x1, x2, x3,..., which occur with probabilities P(X=x), the _____ of X is calculated as E(X)=μ=Σx1P(X=x1)

Poisson Process

An experiment satisfies this if: -The number of successes within a specified time or space interval equals any integer between zero and infinity -The number of successes counted in nonoverlapping intervals are independent -The probability that success occurs in any interval is the same for all intervals of equal size and is proportional to the size of the interval

Discrete Random Variable

Assumes a countable number of distinct values.

Continuous Random Variable

Characterized by uncountable values of an interval.

Bernoulli Process

Consists of a series of n independent and identical trials of an experiment such that on each trial: -There are only two possible outcomes, conventionally labeled success and failure; and -Each time the trial is repeated, the probabilities of success and failure remain the same Is a particular type of experiment named after the Swiss mathematician who described it, James Bernoulli (1654-1705)

Risk Neutral

Consumers are ____ if they are indifferent to risk and care only about their expected gains This consumer completely ignores risk and makes his/her decisions solely on the basis of expected gains.

Risk Averse

Consumers are ____ if they care about risk and, if confronted with two choices with the same expected gains, they prefer the one with lower risk. This consumer demands a positive expected gain as compensation for taking risk. This compensation increases with the level of risk taken and the degree of risk aversion.

Poisson Random Variable

Counts the number of occurrences of a certain event over a given interval of time or space

Variance

For a discrete random variable X with values x1, x2, x3, ..., which occur with probabilities P(X=x1), the ____ of X is calculated as Var(X)=σ^2=Σ(x1-μ)^2P(X=x1)

Hypergeometric Random Variable X

For this, the probability of x successes in a random selection of n items is P(X=x)=((S over x)*((N-S) over (n-x))/(N over n) for x=0,1,2,...,n if n<=S or x=0,1,2,...,S if n>S, where N denotes the number of items in the population of which S are successes. *Note: For three parts of formula, see page 163.

Expected Value, Variance, and Standard Deviation of a Binomial Random Variable

If X is a binomial random variable, then E(X)=μ=np Var(X)=σ^2=np(1-p) SD(X)=σ=sqr(np(1-p))

Expected Value, Variance, and Standard Deviation of a Hypergeometric Random Varible

If X is a hypergeometric random variable, then E(X)=μ=n(S/N) Var(X)=σ^2=n(S/N)(1-(S/N))((N-n)/(N-1)) SD(X)=σ=sqr(n*(S/N)(1-(S/N))((N-n)/(N-1))

Binomial Random Variable

Is defined as the number of successes achieved in the n trials of a Bernoulli process. The possible values of a binomial random variable include 0,1, ..., n. The number of successes achieved in the n trials of a Bernoulli process.

Standard Deviation

Of X is SD(X)=σ=sqr(σ^2)

Cumulative Distribution Function

Of X is defined as P(X<=x)

Probability Mass Function

Of a discrete random variable X is a list of the values of X with the associated probabilities, that is, the list of all possible pairs (x, P(X=x)).

Probability of Successes Equation for Binomial Random Variable

P(X=x)=(n on top of x)p^x)(1-p)^(n-x)=(n!/(x!(n-x)!)*(p^x)*(1-p)^(n-x) for x=0,1,2,...,n. By definition, 0!=1. *Note: For two parts of equation, see page 153

Binomial Distribution

Shows the probabilities associated with the possible values of X.

Hypergeometric Distribution

Used in place of the binomial distribution when we are sampling without replacement from a population whose size N is not significantly larger than the sample size n.