Download Lect6 - ProbStat2012

Some Continuous Probability
Distributions
By: Prof. Gevelyn B. Itao
Probability and Statistics
Continuous Uniform Distribution
 The density function of the continuous uniform
random variable X on the interval [A, B] is
Probability and Statistics
Theorem 6.1
The mean and variance of the continuous uniform distribution
f (x; k) are
and
Probability and Statistics
Continuous Uniform Distribution
Example 5.1: Suppose that a large conference room for a
certain company can be reserved for no more than 4
hours. However, the use of the conference room is such
that both long and short conferences occur quite often.
In fact, it can be assumed that length X of a conference
has a uniform distribution on the interval [0, 4].
a. What is the probability density function?
b. What is the probability that any given conference
lasts at least 3 hours?
Probability and Statistics
Continuous Uniform Distribution
Example 5.2: Calculate the mean and variance in prob 1.
Probability and Statistics
Normal Distribution
Properties of Normal Curve
1. The mode,
which is the
point on the
horizontal
axis where
the curve is a
maximum,
occurs at x = 
Probability and Statistics
Normal Distribution
Properties of Normal Curve
2. The curve is
symmetric about
a vertical axis
through the
mean 
Probability and Statistics
Normal Distribution
Properties of Normal Curve
3. The curve has its
points of
inflection at
x =   , is
concave
downward if
 -<X< +,
and is upward
otherwise
Probability and Statistics
Normal Distribution
Properties of Normal Curve
4. The normal curve
approaches the
horizontal axis
asymptotically as
we proceed in
either direction
away from the
mean.
Probability and Statistics
Normal Distribution
Properties of Normal Curve
5. The total area
under the curve
and above the
horizontal axis is
equal to 1.
Probability and Statistics
Areas Under the Normal Curve
Probability and Statistics
Areas Under the Normal Curve
Probability and Statistics
Definition 6.1
Standard Normal Distribution
 The distribution of a normal random variable with
mean zero and variance 1.
Probability and Statistics
Areas Under the Normal Curve
Example 6.3: Given a standard normal distribution, find the area
under the curve that lies
a. to the right of z = 1.84
b. between z = -1.97 and z = 0.86
Probability and Statistics
Areas Under the Normal Curve
Example 6.3: Given a standard normal distribution, find the area
under the curve that lies
a. to the right of z = 1.84
Probability and Statistics
Areas Under the Normal Curve
Example 6.3: Given a standard normal distribution, find the area
under the curve that lies
b. between z = -1.97 and z = 0.86
Probability and Statistics
Areas Under the Normal Curve
Example 6.4: Given a standard normal distribution, find the value
of k such that
a. P (Z > k) = 0.3015
b. P (k < Z < -0.18) = 0.4197
Probability and Statistics
Areas Under the Normal Curve
Example 6.4: Given a standard normal distribution, find the value
of k such that
a. P (Z > k) = 0.3015
From table A.3:
Probability and Statistics
Areas Under the Normal Curve
Example 6.4: Given a standard normal distribution, find the value
of k such that
b. P (k < Z < -0.18) = 0.4197
From table A.3:
Probability and Statistics
Areas Under the Normal Curve
Example 6.5: Given a normal distribution with  = 50 and  = 10,
find the probability that X assumes a value between 45 and 62.
Probability and Statistics
Areas Under the Normal Curve
Example 6.5: Given a normal distribution with  = 50 and  = 10,
find the probability that X assumes a value between 45 and 62.
Probability and Statistics
Areas Under the Normal Curve
Example 6.5: Given a normal distribution with  = 50 and  = 10,
find the probability that X assumes a value between 45 and 62.
Probability and Statistics
Areas Under the Normal Curve
Example 6.6: Given a normal distribution with  = 40 and  = 6,
find the value of x that has
a. 45% of the area to the left
b. 14% of the area to the right
Probability and Statistics
Areas Under the Normal Curve
Example 6.6: Given a normal distribution with  = 40 and  = 6,
find the value of x that has
a. 45% of the area to the left
From table A.3:
Probability and Statistics
Areas Under the Normal Curve
Example 6.6: Given a normal distribution with  = 40 and  = 6,
find the value of x that has
b. 14% of the area to the right
Probability and Statistics
Areas Under the Normal Curve
Example 6.7: A research scientist reports that mice will
live an average of 40 months when their diets are
sharply restricted and then enriched with vitamins and
proteins.
Assuming that the lifetimes of such mice are normally
distributed with a standard deviation of 6.3 months, find
the probability that a given mouse will live
a. more than 32 months;
b. less than 28 months;
c. between 37 and 49 months.
Probability and Statistics
Areas Under the Normal Curve
Example 6.8: The finished inside diameter of a piston ring
is normally distributed with a mean of 10 centimeters
and a standard deviation of 0.03 centimeter.
a. What proportion of rings will have inside diameters
exceeding 10.075 centimeters?
b. What is the probability that a piston ring will have
an inside diameter between 9.97 and 10.03
centimeters?
c. Below what value of inside diameter will 15% of the
piston rings fall?
Probability and Statistics
Areas Under the Normal Curve
Example 6.9: A lawyer commutes daily from his suburban home to his
midtown office. The average time for a one-way trip is 24 minutes,
with a standard deviation of 3.8 minutes. Assume the distribution of
trip times to be normally distributed.
a. What is the probability that a trip will take at least 1/2 hour?
b. If the office opens at 9:00 A.M. and he leaves his house at 8:45
A.M. daily, what percentage of the time is he late for work?
c. If he leaves the house at 8:35 A.M. and coffee is served at the
office from 8:50 A.M. until 9:00 A.M., what is the probability that
he misses coffee?
d. Find the length of time above which we find the slowest 15% of
the trips.
e. Find the probability that 2 of the next 3 trips will take at least 1/2
hour.
Probability and Statistics
Normal Approximation to the Binomial
Theorem 6.2
If X is a binomial random variable with mean  = np and variance
2 = npq, then the limiting form of the distribution of
as n  , is the standard normal distribution n (z; 0,1)
Probability and Statistics
Normal Approximation to the Binomial
Example 6.10: A process for manufacturing an electronic
component is 1% defective. A quality control plan is to
select 100 items from the process, and if none are
defective, the process continues. Use the normal
approximation to the binomial to find
a. the probability that the process continues for the
sampling plan described;
b. the probability that the process continues even if the
process has gone bad (i.e., if the frequency of
defective components has shifted to 5.0% defective).
Probability and Statistics
Normal Approximation to the Binomial
Example 6.11: A process yields 10% defective items. If 100
items are randomly selected from the process, what is the
probability that the number of defectives
a. exceeds 13?
b. is less than 8?
Probability and Statistics
Gamma Distribution
 The continuous random variable X has a gamma
distribution, with parameters α and β, if its
density function is given by
where α > 0 and β > 0.
Probability and Statistics
Definition 6.2
The gamma function is defined by
where α > 0.
Probability and Statistics
Theorem 6.3
The mean and variance of the gamma distribution are
µ = αβ
and
σ2 = αβ2
Probability and Statistics
Normal Approximation to the Binomial
Example 6.12: In a certain city, the daily consumption of
electric power, in millions of kilowatt-hours, is a random
variable X having a gamma distribution with mean µ = 6
and variance σ2 = 12.
a. Find the values of α and β.
b. Find the probability that on any given day the daily
power consumption will exceed 12 million
kilowatthours.
Probability and Statistics
Normal Approximation to the Binomial
Example 6.13: Suppose that the time, in hours, taken to
repair a heat pump is a random variable X having a
gamma distribution with parameters α =2 and β =1/2.
What is the probability that the next service call will
require
a. at most 1 hour to repair the heat pump?
b. at least 2 hours to repair the heat pump?
Probability and Statistics
Exponential Distribution
 The continuous random variable X has an
exponential distribution, with parameter β, if
its density function is given by
where β > 0.
Probability and Statistics
Theorem 6.4
The mean and variance of the exponential distribution are
µ=β
and
σ2 = β2
Probability and Statistics
Normal Approximation to the Binomial
Example 6.14: The life, in years, of a certain type of
electrical switch has an exponential distribution with an
average life β = 2. If 100 of these switches are installed
in different systems, what is the probability that at most
30 fail during the first year?