Probability Density Functions

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Probability Density Functions
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Overview

• To frame our discussion, consider

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Outline
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Continuous Sample Space

• The concept of discrete random variable and its
  associated probability distribution is extended
  to continuous sample spaces. We can describe
  continuous random variables and connect with
  these probability density functions.

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Probability Distribution
Let X be a continuous rv. Then a probability
distribution or probability density function
(pdf) of X is a function f (x) such that for any
two numbers a and b,
The graph of f is the density curve.
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• f is an integrable real-valued function satisfying

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Probability Density Function
is given by the area of the
shaded region.
b
a
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Uniform Distribution
A continuous rv X is said to have a uniform
distribution on the interval A, B if the pdf of
X is
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Probability for a Continuous rv
If X is a continuous rv, then for any number c,
P(x c) 0. For any two numbers a and b with a
lt b,
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The Cumulative Distribution Function
The cumulative distribution function, F(x) for a
continuous rv X is defined for every number x by
For each x, F(x) is the area under the density
curve to the left of x.
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Using F(x) to Compute Probabilities
Let X be a continuous rv with pdf f(x) and cdf
F(x). Then for any number a,
and for any numbers a and b with a lt b,
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Obtaining f(x) from F(x)
If X is a continuous rv with pdf f(x) and cdf
F(x), then at every number x for which the
derivative
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Percentiles
Let p be a number between 0 and 1. The (100p)th
percentile of the distribution of a continuous rv
X denoted by , is defined by
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Median
The median of a continuous distribution, denoted
by , is the 50th percentile. So
satisfies That is, half
the area under the density curve is to the left
of
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Expected Value
The expected or mean value of a continuous rv X
with pdf f (x) is
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Expected Value of h(X)
If X is a continuous rv with pdf f(x) and h(x) is
any function of X, then
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Variance and Standard Deviation
The variance of continuous rv X with pdf f(x) and
mean is
The standard deviation is
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Expected Value of h(X)
If X is a continuous rv with pdf f(x) and h(x) is
any function of X, then
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Normal Distributions
A continuous rv X is said to have a normal
distribution with parameters
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Normal Distribution

• The normal distribution is a family of
  probability distributions. A normal distribution
  is specified by the mean and variance (sometimes
  written as N(m,s2)).

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Standard Normal Distributions
The normal distribution with parameter values
is called a standard
normal distribution. The random variable is
denoted by Z. The pdf is
The cdf is
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Standard Normal

• N(0,1)
• Any normal distribution becomes a standard normal
  distribution with the transformation
• Z is the standardized random variable

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z-scores

• We can convert the x-values provided from the
  original data to what are called z-scores.
• Notice the z-score tells us the number of
  standard deviations an x-value is from the
  original mean.

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Standard Normal Distribution
Let Z be the standard normal variable. Find
(from table)
a.
Area to the left of 0.85 0.8023
b. P(Z gt 1.32)
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Find the area to the left of 1.78 then subtract
the area to the left of 2.1.
0.9625 0.0179
0.9446
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Notation
will denote the value on the measurement
axis for which the area under the z curve lies to
the right of
0
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Ex. Let Z be the standard normal variable. Find
z if a. P(Z lt z) 0.9278.
Look at the table and find an entry 0.9278 then
read back to find
z 1.46.
b. P(z lt Z lt z) 0.8132
P(z lt Z lt z ) 2P(0 lt Z lt z)
2P(z lt Z ) ½
2P(z lt Z ) 1
0.8132
P(z lt Z ) 0.9066
z 1.32
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Normal Curve
Approximate percentage of area within given
standard deviations (empirical rule).
99.7
95
68
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Ex. Let X be a normal random variable with

0.2266
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Ex. A particular rash shown up at an elementary
school. It has been determined that the length
of time that the rash will last is normally
distributed with
Find the probability that for a student selected
at random, the rash will last for between 3.75
and 9 days.
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0.9772 0.0668
0.9104
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Normal Approximation to the Binomial Distribution
Let X be a binomial rv based on n trials, each
with probability of success p. If the binomial
probability histogram is not too skewed, X may be
approximated by a normal distribution with
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Ex. At a particular small college the pass rate
of Intermediate Algebra is 72. If 500 students
enroll in a semester determine the probability
that at least 375 students pass.
0.9394
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Excel Functions

• NORMDIST(x,mean,standard deviation,cumulative)
• NORMINV(probability,mean,standard deviation)
• NORMSDIST(z score)
• NORMSINV(probability)

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Type 1

• P(altXltb)P(Xltb)-P(altX)F(b)-F(a)
• The effort, in minutes, required for a developer
  to repair a Type I error is a normally
  distributed random variable, N(100,25). What is
  the probability a person takes between 90 and 120
  minutes to repair Type I error?

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Type 2

• Using the z-score associated with the
  probability, solve for µ.
• Given a normally distributed random variable X
  with variance 225, what should the mean of this
  random variable be so only 3 of the values fall
  above 65?

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Problem

• Company records indicate that the time an
  individual spends preparing for a code inspection
  is normally distributed with a mean of 55 minutes
  and a standard deviation of 15 minutes.
• What is the probability an employee spends more
  than 75 minutes preparing for a review?

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Problem

• The committee assigned to developed a
  certification process for test engineers
  collected data on an examination that
  constructed. The scores follow a normal
  distribution with a mean of 72 and a standard
  deviation of 18. The committee decides that only
  the top 10 of the scores on the exam should be
  considered passing.
• What is the passing score?

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Chebyshevs Theorem

• If a probability distribution has a mean m and a
  standard deviation s, the probability of getting
  a value which deviates from m by at least ks is
  at most 1/k2.