STA220H5 Chapter Notes - Chapter 4.4: Binomial Distribution, Random Variable, Standard Deviation
4.4-4.6
Characteristics of a binomial random variable
1. The experiment consists of n identical trials
2. There are only two possible outcomes on each trial. We will denote one
outcome by S (for success) and the other by F (for failure).
3. The probability of S remains the same from trial to trial. This probability is
denoted by p, and the probability of F is denoted by q = 1 – p.
4. The trials are independent.
5. The binomial random variable x is the number of S’s in n trials.
The binomial probability distribution
P (x) = (n x) p x q n-x x = ,,… n
Where
P=probability of a success on a single trial
Q=1-p
N=number of trials
X=number of successes in n trials
N-x= number of failures in n trials
(N x) =n! /x! (N-x)!
Mean, variance, and standard deviation for a binomial random variable
Mean: u =np
Variance: a2 =npq
Standard deviation: a=npq
Characteristics of a poisson random variable
1. The experiment consists of counting the number of times a certain event
occurs during a given unit of time or in a given area or volume (or weight,
distance, or any other unit of measurement).
2. The probability that an event occurs in a given unit of time, area, or volume is
the same for all the units.
3. The number of events that occur in one unit of time, area, or volume is
independent of the number that occur in other units.
4. The mean (or expected) number of events in each unit is denoted by the
Greek letter lambda (y).
Probability distribution, mean, and variance for a poisson random variable
P (x) =yxe-y/x! X = ,,… u=y a=y
Where
Y = mean number of events during a given unit of time, area, volume, etc.
E= 2.71828
Characteristics of a hypergeometric random variable
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Document Summary
Characteristics of a binomial random variable: the experiment consists of n identical trials, there are only two possible outcomes on each trial. We will denote one: the trials are independent. outcome by s (for success) and the other by f (for failure), the probability of s remains the same from trial to trial. This probability is denoted by p, and the probability of f is denoted by q = 1 p: the binomial random variable x is the number of s"s in n trials. P (x) = (n x) p x q n-x (cid:523)x = (cid:882),(cid:883),(cid:884) n(cid:524) P=probability of a success on a single trial. N-x= number of failures in n trials (n x) =n! Mean, variance, and standard deviation for a binomial random variable. Probability distribution, mean, and variance for a poisson random variable. P (x) =yxe-y/x! (cid:523)x = (cid:882),(cid:883),(cid:884) (cid:524) u=y a(cid:884)=y. Y = mean number of events during a given unit of time, area, volume, etc.
