Chapter 5 Discrete Random Variables

Random Variable

quantitative variable whose value depends on chance

Discrete Random Variable

a random variable whose possible values can be listed

Random Variable Notation

• lower case x,y,z to denote variables…

• so to denote the random variable we use X,Y,Z

• {X=x} denotes the event that the random variable X equals x

• P(X=x) denotes the probability that the random variable X equals x

Sum of the Probabilities of a Discrete Random Variable

For any discrete random variable X,

we have sum of all P(X=x)=1

Interpretation of a Probability Distribution

In a large number of independent observations of a random variable X, the proportion of times each possible value occurs will approximate the probability distribution of X; or, equivalently, the proportion histogram will approximate the probability histogram for X.

Simulating Probability Distribution

Simulating a random variable means that we use a computer or statistical calculator to generate observations of the random variable

Mean of a Discrete Random Variable (expected value, expectation)

sum of all xP(X=x)

Interpretation of the Mean of a Random Variable

• in a large number of independent observations of a random variable X, the average value of those observations will approximately equal the mean, of X

• the larger the number of observations, the closer the average tends to be to the mean

The Binomial Distribution

a type of frequency dist used when there are two exact outcomes from each trials (success or failure)

Binomial Coefficients

If n is a positive integer and x is a nonnegative integer less than or equal to n, then the binomial coefficient (n/x) is defined as (n/x)=n!/x!(n-x)!

Bernoulli Trials

1) the experiment (each trial) has two possible outcomes, denoted generically s, for success, and f, for failure

2) the trials are independent, meaning that the outcome on one trials in no way affects the outcome on other trials

3) the probability of a success, called the success probability and denoted, remains the same from trial to trial

Number of Outcomes Containing a specified Number of successes

In n Bernoulli trials, the number of outcomes that contains exactly x success equals the binomial coefficient (n/x)

p(X=x)=(n/x)P^x(1-p)^n-x, x=0,1,2,…n

To Find a Binomial Probability (Assumptions 4 of them)

• n trials are to be performed

• two outcomes, success or failure, are possible for each trial

• the trials are independent

• the success probability, p, remains the same from trial to trial

Steps to find a Binomial Probability (4)

1) Identify success

2) Determine p, the success probability

3) Determine n, the number of trials

4) the binomial probability formula

Mean of Binomial Random Variable

Mean=np

Standard Deviation of a Binomial Random Variable

small sigma= square root of (np(1-p)