Chapter 7: Random Variables and Discrete Probability Distributions Flashcards
Random variable
A function or a rule that assigns a number to each outcome of an experiment
Name of random variable = uppercase letter
Value of random variable = lowercase letter
Probability X has the value of x = P(X= x) or just P(x)
Discrete random variable
Random variable that can take on a countable number of values
Continuous random variable
Random variable whose values are uncountable (could be anything - infinitely divisible)
Probability distribution
A table, formula, or graph that describes the values of a random variable and the probability associated with these values
When multiple outcomes are possible to reach a single value then the sum of these possible outcomes is the probability of that value
Probability distribution of a sample represents the probability distribution of the population
Requirements for a distribution of a discrete random variable
1) probability of x is between 0 and 1 for all values of x
2) some of the probability of all possible values of x = 1
Population mean
Aka expected value (of the random variable) E(X)
Weighted average of all values
Weight = probability
E(X) = sum of all individual values of x multiplied by their probability
Population variance
The weighted average of all the squared deviations from the mean
So sum of all difference between value of x and mean squared times the probability of that value of x
Shortcut:
Sum of all (values of x squared multiplied by the probability of x) less the mean squared
Laws of expected value
Where X is the random variable and c is a constant
E(c) = c E(X+c) = E(X)+c E(cX) = cE(X)
Laws of variance
Where X is the random variable and c is a constant
V(c)=0
V(X+c)= V(X)
V(cX) =c^2V(X)
Binomial experiment
- consists of a fixed number of trials (number of trials = n)
- each trial has two possible outcomes, one labeled success other labeled failure
- probability of success is p. Probability of failure is 1-p
- trials are independent (outcome of one trial does not affect the outcome of others)
Bernoulli Process
- each trial has two possible outcomes, one labeled success other labeled failure
- probability of success is p. Probability of failure is 1-p
- trials are independent (outcome of one trial does not affect the outcome of others)
Binomial experiment without fixed number of trials
Binomial random variable
Random variable is the number of successes in the n trials
A discrete variable (only integers)
Binomial probability distribution
The probability of x successes in a binomial experiment with n trials (where the probability of success = p)
P(x)= (n!/ x!(n-x)!)p^x(1-p)^(n-x)
Remember 0!=1
(First half determines number of ways to get that number of successes second half determines probability of each way)
Cumulative probability
The probability that a random variable is less than or equal to a value
For value x finding P(X<=x)
Sum of probabilities of all random variable values x and below
Binomial table
Provides cumulative binomial probabilities for given values of n (tests) and p (probability of success in a single experiment)
Using binomial table to find the binomial probability of P(X>=x)
Basically using the compliment rule. The probability that X is greater than or equal to x value is 1 minus the probability it is less than or equal to x-1 so:
P(X>=x) = 1 - P(X<=[x-1])
Using the binomial table to find the binomial probability or P(X=x)
P(X=x) = P(X<=x) - P(X<=[x-1])
Binomial probability in excel
=BINOMDIST([x], [n], [p], True or False)
x = number of successes n = total number of tests p = probability of success True = cumulative probability (P(X<=x) False = probability of individual value x
Mean of a binomial random variable
= n * p
Number of tests * probability of success
Variance of a binomial random variable
= np(1-p)
Number of tests * probability of success * probability of failure
Standard distribution of a binomial random variable
Square root of variance
=√np(1-p)
Poisson random variable
The number of successes that occur in a period of time or an interval of space in a Poisson experiment
Poisson experiment
Characteristics:
- number of success that occur in any interval is independent of the number of successes that occur in any other interval
- the probability of a success in an interval is the same for all equal sized intervals
- the probability of a success in an interval is proportional to the size of the interval
- the probability of more than one success in an interval approaches 0 as the interval becomes smaller
Poisson probability distribution
A discrete probability distribution
probability of x successes in a specific interval = e to the negative power of the mean number of successes times the mean number of successes to the power of x all divided by x factorial
P(x) = ((e^- mean)*(mean^x))/x!
e= base of the natural logarithm
Variance of a Poisson random variable
Equal to the mean of the random variable
Calculating the Poisson probability distribution when the interval does not match
- find the proportion of the interval of interest to the given mean interval
- multiply or divide the mean by the relative proportion
- solve qsbusyal
Cumulative Poisson probabilities by formula
Find each of the discrete probability and sun
Poisson table
Shows cumulative Poisson probabilities for selected mean values
Gives values for P(x)<=x
Using Poisson table to find Poisson probability P(X>=x)
P(X>=x)= 1 - P(X<=[x-1])
Using Poisson probability table to find an individual value of x
P(X=x) = P(X<=x) - P(X<=[x-1])
Poisson probability in excel
=POISSON([x],[mean], True or False
True = cumulative probability, false = individual
