Chapter 5 - Discrete Probability Distribution p.219 Flashcards
Binomial experiment p.244
An experiment having four properties
1. The experiment consists of a sequence of n identical trials.
2. Two outcomes are possible on each trial. We refer to one outcome as a success and the other outcome as a failure.
3. The probability of a success, denoted by p, does not change from trial to trial. Consequently, the probability of a failure, denoted by 1 - p, does not change from trial to trial.
The trials are independent.
Binomial probability distribution p.244
A probability distribution showing the probability of x successes in n trials of a binomial experiment.
Binomial probability function p.248
The function used to compute binomial probabilities.
Bivariate probability distribution p.234
A probability distribution involving two random variables. A discrete bivariate probability distribution provides a probability for each pair of values that may occur for the two random variables.
Continuous random variable p.222
A random variable that may assume any numerical value in an interval or collection of intervals.
Discrete random variable p.221
A random variable that may assume any numerical value in an internal or collection of intervals.
Discrete uniform probability distribution p.226
A probability distribution for which each possible value of the random variable has the same probability.
Empirical discrete distribution p.224
A discrete probability distribution showing the probability of x successes in n trials from a population with r successes and N - r failures.
Expected value p.229
A measure of the central location of a random variable.
Hypergeometric probability distribution p.258
- A probability distribution showing the probability of x successes in n trials from a population with r successes and N - r failures.
- Closely related to the binomial distribution. The two probability distributions differ in two key ways:
1. With the hypergeometric distribution, the trials are not independent
2. And the probability of success changes from trial to trial
Hypergeometric probability function p.258
- The function used to compute hypergeometric probabilities.
- Used to compute the probability that in a random selection of n elements. Selected without replacement, we obtain x elements labeled success and n - x elements labeled failure.
- f(x), the probability of obtaining x successes in n trials.
Poisson probability distribution p.254
A probability distribution showing the probability of x successes in n trials of a binomial experiment.
Poisson probability function p.254
The function used to compute binomial probabilities.
Probability distribution p.224
A description of how the probabilities are distributed over the values of the random variable.
Probability function p.224
A function, denoted by f(x), that provides the probability that x assumes a particular value for a discrete random variable.
Standard deviation p.230
The positive square root of the variance.
Variance p.229
A measure of the variability, or dispersion, of a random variable.
5.3 Discrete Uniform Probability Function
- f(x) = 1/n
- Where
- N = the number of values the random variable may assume.
5.4 Expected Value of a Discrete Random Variable
E(x) = u = Sum(x*f(x))
5.5 Variance of a Discrete Random Variable
Var(x) = o2 = Sum((x - u)2 * f(x))
5.6 Covariance of Random Variables x and y
The covariance and/or correlation coefficient are good measures of association between two random variables.
Covariance(x,y) = [Var(x + y) - Var(x) - Var(y)]/2
5.7 Correlation between Random Variables x and y
To get a better sense of the strength of the relationship between two random variables we can compute the correlation coefficient.
Correlation(x,y) = Covariance(x,y)/(StDev(x) * StDev(y))
5.8 Expected Value of a Linear Combination of Random Variables x and y
E(ax + by) = aE(x) + bE(y)
- Where ‘a’ represents the coefficient of x and ‘b’ represents the coefficient of y in the linear combination.
5.9 Variance of a Linear Combination of Two Random Variables
When the covariance between two random variables is known, we can compute the variance of a linear combination of two variables.
Var(ax + by) = a2Var(x) + b2Var(y) + (2ab)Covariance(x,y)
5.10 Number of Experimental Outcomes Providing Exactly x Successes in n Trials
(n . x) = n!/(x!(n - x)!)
Where, n! = n(n - 1)(n - 2) . . . (2)(1)
And, by definition, 0! = 1
5.12 Binomial Probability Function
f(x) = (n . x)px(1 - p)(n - x)
Where
- X = the number of successes
- P = the probability of a success on one trial
- N = the number of trials
- f(x) = the probability of x successes in n trials
- (n . x) = n!/(x!(n - x)!)
5.16 Hypergeometric Probability Function
f(x) = (r . x)((n - r) . (N - x))/(N . n)
Where
- x = the number of successes
- n = the number of trials
- f(x) = the probability of x successes in n trials
- N = the number of elements in the population
- r = the number of elements in the population labeled success
5.17 Expected Value for the Hypergeometric Distribution
E(x) = u = n(r/N)
5.18 Variance for the Hypergeometric Distribution
Var(x) = StandardDeviation^2 = o^2 = n(r/N)(1 - r/N)((N - n)/(N - 1))
Random Variable
A random variable is a numerical description of the outcome of an experiment.
Required conditions for a discrete probability function
f(x) >= 0
Sum(f(x)) = 1
Properties of a Poisson Experiment
- The probability of an occurrence is the same for any two intervals of equal length.
- The occurrence or nonoccurrence in any internal is independent of the occurrence or nonoccurrence in any other interval.
