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Continuous Probability Distribution




Lecture 6 Random Variables

 

Definition

A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.

For example define the random variable X as the number of heads in 2 tosses of a fair, 50-50 coin. The sample space is the corresponding outcomes in this sample space get associated with values of the random variable X as because the outcomes have 1,2,1, and 0 heads respectively.

Two Types of Random Variables

Ø Random Variable:

Discrete random variable

Continuous random variable

 

Discrete Random variable

A discrete random variable X has a countable number of possible values.

The following are examples of discrete random varibles:

1. the number of seizers an epileptic patient has in a given week: x=0,1,2,…

2. The number of voters in a sample of 500 who favor impeachment of the president: x=0,1,2,…,500

3. The number of students applying to medical schools this year: x=0,1,2,…

4. The number of errors on a page of an account’s ledger: x=0,1,2,…

5. The number of customers waiting to be served in a restaurant at a particular time: x=0,1,2,…

Probability Distributions for Discrete Random Variables

The probability distribution of X lists the values and their probabilities:

Value of X Probability
X1 p1
X2 p2
X3 p3
: :
: :
Xk pk

The Probabilities pi must satisfy two requirements:

1. Every probability pi is a number between 0 and 1.

2. p1+p2 … +pk = 1

We usually summarize all the information about a random variable with a probability table like:

X 0 1 2------------------------------------P(x) 1/4 1/2 1/4

this is the probability table representing the random variable X defined above for the 2 toss coin tossing experiment. There is one outcome with zero heads, 2 with one head, and one with 2 heads. All outcomes are equally likely, and this means the probabilities are defined as the number of outcomes in the event divided by the total number of outcomes.

 

Definition

The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume.

Continuous Probability Distribution

Continuous Random variable

Random variables that can assume values corresponding to any of the points contained in one or more intervals are called continuous. A continuous random variable takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.





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