Random Variables and Probability Distributions

Section 4 – Random Variables
and Probability Distributions
Random Variables
• Capital X denotes a random variable
– Based on the distribution, there are different
probabilities for each value that X could take on
– We’re familiar with the normal distribution
already, but we’ll learn about distributions with
other “shapes” later
Discrete vs. Continous
• Discrete RV: X can only be from a finite set of
values (usually integers or only some integers)
– Ex: The number of customers that walks in
– Probability Mass Function (pmf)
• Continuous RV: X can be a value anywhere in
a range of numbers
– Ex: The exact time a customer walks in
– Probability Density Function (pdf)
Discrete RV’s & Distributions
• Probability function of a discrete RV
– Called probability mass function (pmf)
– See graph (Actex page 112)
0  p(x)  1
 p(x)  1
P(X  A) 
 p(x)  P(A)
x A
Continuous RV’s & Distributions
• Probability function of a continuous RV
– Called a probability density function (pdf)
– See graph (Actex p113)
f(x) = probability density function (pdf)
F(x) = P(X<=x) = cumulative probability density
function (cdf)
S(x) = 1- F(x) = P(X>x)
Continuous RV’s & Distributions
• Properties
f (x)  0
 f (x)dx  1
P[X  (a,b)]  P(a  X  b) 
f (x)dx
F(b)  P(  x  b) 
f (x)dx  F(b)  F(a)
f (x)dx
Greater Than vs. Greater Than or Equal To
• Discrete
• The distinction is important (there is likely some
probability on the integer included or left out)
• Continuous
• The distinction is NOT important
• For any value x, f(x) = 0
• No one point has any probability (integrating from a to a is
0) so there will be no probability left out by not including
the “edge”
Mixed Distributions
• Mixed Distribution: A random variable’s
distribution where it has discrete probabilities for
some interval and continuous on another
– Basically continous and discrete
– f(x) is a piecewise function
– Total probability still sums to 1
• Note: these are not “mixtures of distributions” –
that topic will come much later, but remember to
study both!
Conditional Distributions
• Recall:
• Suppose :
– fX(x) is “marginal density” of X
f X (x)
– A is an event
f X |A (x | A) 
– We will go into more detail about calculating the
marginal density
in our next review session.
Note: if X is not an outcome in event A, fX|A(x|A)=0