Download Random Variables

Random Variables
Discrete Random
Variables:
a random variable
that can assume only
a countable number
of values. The value
of a discrete random
variable comes from
counting.
Continuous Random
Variable:
random variables that
can assume any value
on a continuum.
Measurement is
required to determine
the value for a
continuous random
variable.
Continuous Probability
Distributions
The probability distribution of a
continuous random variable is
represented by a probability
density function that defines a
curve. The area under the curve
corresponds to the probabilities for
the random variable.
Continuous Probability
Distributions
The Continuous Uniform Distribution
A probability distribution in which the
probability of a value occurring between two
points, a and b, is the same as the probability
between any other two points, c and d, given
that the distance between a and b is equal to
the distance between c and d.
f (x) = 1 / ( b - a ) if a < x < b
The Uniform Probability
Density Function
Mean and standard deviation of the uniform
probability density function:
a b

2
b a
 
12
EXAMPLE
Suppose the research department of a
steel manufacturer believes that one of
the company’s rolling machines is
producing sheets of steel of varying
thickness. The thickness is a uniform
random variable with values between 150
and 200 millimeters. Any sheet less than
160 millimeters must be scrapped because
they are unacceptable to buyers.
EXAMPLE
Calculate the mean and standard
deviation of x, the thickness of the sheets
produced by this machine. Then graph
the probability distribution and show the
mean on the horizontal axis.
Calculate the fraction of steel sheets
produced by this machine that have to be
scrapped.
Continuous Probability
Distributions
The Normal Distribution
A bell-shaped, continuous distribution with the
following properties:
It is unimodal; the normal distribution peaks at a single
value.
It is symmetrical; 50% of the area under the curve lies
left of the center and 50% lies right of the center.
The mean, mode, and median are equal.
It is asymptotic; the normal distribution approaches the
horizontal axis on each side of the mean toward + 
The Normal Distribution
The Normal Distribution is defined by
two parameters:
 X  Mean
  Variance
2
X
The Standard Normal
Distribution
The Standard Normal Distribution
is a continuous, symmetrical, bellshaped distribution that has a
mean of 0 and a standard
deviation of 1.
The Z Score
The Z Score is the number of standard
deviations between the mean and the
point X.
Z
X  X
X