STAT-S 300 Lecture Notes - Lecture 13: Random Variable, Probability Distribution, Standard Deviation
6.1 Notes- Discrete and Continuous Random Variables 10-9-13
• Probability model- describes possible outcomes of chance process and likelihood that
each outcome occurs
• Random variable- takes numerical values that describe outcomes of some chance
process
o Discrete
o Continuous
• Probability distribution of random variable gives its possible values and their
probabilities
• Discrete Random Variables
o Discrete random variable X takes a set of possible fixed values with gaps
between (can be infinitely many fixed values)
▪ Probability of a discrete random variable X lists the values xi and their
probabilities pi
▪ Probabilities pi must satisfy 2 requirements
• Every probability pi is number between 0 and 1
• Sum of probabilities is 1
▪ Ex. X = number of rolls on fair die until you get a 6- discrete with
infinite possible values
▪ Ex. X = shoe size- discrete with finite possible values; shows gaps
between values
o To find probability of any event, add probabilities pi of the particular values xi
that make up the event
• Mean (Expected Value) of a Discrete Random Variable
o Mean (expected value) of X
▪ Represented by or E(X)
▪ Suppose X is discrete random variable with probability distribution below
Value
x1
x2
x3
…
Probability
p1
p2
p3
…
▪ To find mean of X, multiply each possible value by its probability, then
add all products:
▪ Not always actual value in set
▪ Must show work when calculating
▪ Interpret as a “long run average”
• Standard Deviation and Variance of a Discrete Random Variable
o Variance of random variable similar to definition of variance for
quantitative data
▪ Average of squared deviation (xi – x)2 of values of variable X from its
mean
▪ Suppose X is discrete random variable with probability distribution below
and is mean of X
Value
x1
x2
x3
…
Probability
p1
p2
p3
…
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