10. Binomial Distribution

A ________ __________ is a variable (typically represented by X, or another capital letter) that takes on a different value for each outcome of a random phenomenon, and is determined by chance

random variable

A _______ variable is a variable whose value is a numerical outcome of a random phenomenon

random

The two broad types of random variables are _______ random variable and __________ random variable

discrete

continuous

A ______ random variable is a random variable that has a countable set of distinct values

Examples:

→ X = number of heads from a coin toss

→ Y = the result of a single die roll

→ the number of smokers in a random sample

discrete

A _________ random variable is a random variable that has infinitely many values. A variable is _________ if between any two possible values of the variable, there exists another possible value for the variable

Examples:

→ X = height

→ Y = serum creatinine level

→ Z = volume

continuous

A __________ __________ is a graph, table, or formula that gives the probability of a given outcomes’s occurance

probability distribution

A _________ ___________ ________ (pdf) is any mathematical relationship between each possible value and the probability of that value

P(x) = P(X = x)

probability distribution function

Requirements for a probability distribution:

1) All of the individual probabilities must be between ___ and ___ (, inclusive);

2) The sum of all probabilities must be equal to _

0 and 1

1

When dealing with probability distributions, a _______ is the set of all possible outcomes for a random variable

→ it refers to a hypothetical ________ of numbers, not a ________ of people

population

_________ ________ have shapes, centers, and variability

probability distributions

The _____, or expected value, of any discrete random variable X represents the average value of the outcomes

(each observed X value is multiplied by its respective probability)

mean

A __________ random variable is a particular type of discrete random variable with the following characteristics:

→ There is a ______ number of identical trials

→ There are only ___ possible outcomes for each trial. One success (S) and one failure (F).

→ The probability of S remains the _____ from trial to trial (denoted as p)

→The outcomes of the trials are __________

binomial

fixed

two

same

independent

When working in a binomial setting, the number of successes X out of n trial has a binomial distribution with parameters n and p

→ the parameter _ is the total number of trials

→ the parameter _ is the probability of a success on any one observation (the possible values of X are the whole numbers from 0 to n | the choice of what we call a success is ______)

n

p

arbitrary

In order to work with binomial random variables, we must first define the _______ _______ which is the number of ways we can choose k objects from a total of n objects

binomial coefficient

we usually think of the ________ ________ as the number of ways we can arrange k successes among n observations

binomial coefficient

x! is called

x factorial

( n ) =

  k

n!/k!(n-k)!

x! = 1 × 2 × 3 x . . . x ___

x

2! =

1 × 2 = 2

(7 - 3)! =

4! = 1 × 2 × 3 × 4 = 24

0! =

1

Consider 5 letters (A, B, C, D, E). How many different combinations of 3 letters can we choose?

( 5 ) = 5!/3!(5-3)!

3

How to calculate binomial coefficient on calculator?

top number input

press nCr

bottom number input

ENTER

( n ) can be denoted as…

k

C_kⁿ or ⁿC_k or _nC_k

A __________ ________ ________ (pdf) is any mathematical relationship between each possible value and the probability of that value

→ the pdf for a binomial random variable is:

______________________________________

where n= # of trials, x = # of successes, and p = is the probability of success

Shorthand to denote a binomial random variable: X~Bin(n, p), where n = # of trials and p = the probability of a success on any one trial

→ X~Bin(10, 0.5) signifies that X is a binomial random variable with n = _ trials and p = __ probability of success on any one trial

10

0.5

Since there are a fixed number of trials n, when you are interested in the probability of x successes, the number of failures is automatically determined to be __ - __

→ The probability of success is the same for each of the x successes, and the probability of failure is the same for each of the __ - __ failures

n (# of fixed trials) - x (number of specific number of successes in n trials)

Since all the n trials are __________, the probability of the intersection of any single combination of events is simply the product of their corresponding probabilities

→ However, there are more than just the one way to choose which trials are successes and which are failures . . . “n chooses x” different ways

independent

The complement of getting at least one item of a particular type is you get __ items of that type

no