STT 1600 CHAPTER 8

random variable

a letter which is assigned a numerical value depending on the outcome of an experiment.

types of random variables

• discrete random variables

• continuous random variables

• random variables also belong to “families “ in the sense that members of the same RV family share certain characteristics

discrete random variable

a random variable that may take a finite number of values

continuous random variable

a random variable that always takes an infinite number of values within a specified interval of values

Probability notation for a discrete RV

P[X=k] - if X is a discrete RV, and k is a specific number that X could take, then this is the probability that X takes the value of k when the experiment is performed.

The probability distribution function of a discrete random variable (discrete pdf)

• the way that you assign probabilities to each possible value of a discrete RV

• may take the form of a table or an equation

• tells you how likely each outcome is

Conditions for probabilities for discrete RVs

1. every probability must be a number between 0 and 1, inclusive

2. the sum of all non- zero probabilities must be EXACTLY 1.

Using the sample space to find probabilities for discrete RVs

1. make a tree diagram- list all sample points in S

2. identify the value of X for each sample point in S ( calculate the branch probabilities of X)

3. Identify each sample point that leads to a value of X=k, then P[X=k or x] is simply the sum of the probabilities in step 3.

4. make pdf table using branch probabilities

graphing the pdf for a discrete RV

1. X goes on horizontal axis

2. corresponding branch probabilities go on the y axis

3. calculate the area HW ( highest point on histogram, corresponding x value )

The cumulative distribution of a discrete random variable

• P[X less than or equal to k] for some RV X

• Add up all the probabilities where H is less than or equal to k

expected value of an RV

• the mean value of the RV after a very long (infinite ) run of the experiment that produces values of X.

• E (X)= SUM [x * p(x) ] or SUM xip(xi)

• value is read as mew ( population mean )

Standard deviation for a discrete RV

• V (X) = SUM (X- u)² * p(x)

• tells you how spread out the values of X are around the mean

• tells how much variation the random variable typically has