Download Continuous Distributions Notes

Continuous Distributions Notes
Uniform Distribution –
Continuous random variables –
Formulas:
Density curves:
The Citrus Sugar Company packs sugar in bags
labeled 5 pounds. However, the packaging isn’t
perfect and the actual weights are uniformly
distributed with a mean of 4.98 pounds and a range
of .12 pounds.
a) Construct the uniform distribution above.
Unusual density curves:
b) What is the probability that a randomly selected
bag will weigh more than 4.97 pounds?
Example:
1) P(x<2) =
2) P(X=2) =
.5
c) Find the probability that a randomly selected bag
weighs between 4.93 and 5.03 pounds.
.25
3) P(X<2) =
4) P(x > 3) =
1
2
3
4
5
The time it takes for students to drive to school is
evenly distributed with a minimum of 5 minutes and
a range of 35 minutes.
a) Draw the distribution
5) P(1 < x < 3) =
b) What is the probability that it takes less than 20
minutes to drive to school?
1) P(X > 1 =
0.50
2) P(0.5 < x < 1.5) =
0.25
c) What is the mean and standard deviation of this
distribution?
1
2
3
4
Normal Distributions
Strategies for finding probabilities or proportions
in normal distributions
The lifetime of a certain type of battery is normally
distributed with a mean of 200 hours and a
standard deviation of 15 hours. What proportion of
these batteries can be expected to last less than
220 hours?
Do these two normal curves have the same mean?
If so, what is it?
Which normal curve has a standard deviation of 3?
Which normal curve has a standard deviation of 1?
What proportion of these batteries can be
expected to last more than 220 hours?
Empirical Rule:
How long must a battery last to be in the top 5%?
Suppose that the height of male students at PWSH
is normally distributed with a mean of 71 inches and
standard deviation of 2.5 inches. What is the
probability that the height of a randomly selected
male student is more than 73.5 inches?
Standard Normal Density Curves
The heights of the female students at PWSH are
normally distributed with a mean of 65 inches.
What is the standard deviation of this distribution
if 18.5% of the female students are shorter than
63 inches?
The heights of female teachers at PWSH are
normally distributed with mean of 65.5 inches and
standard deviation of 2.25 inches. The heights of
male teachers are normally distributed with mean of
70 inches and standard deviation of 2.5 inches.
Describe the distribution of differences of heights
(male – female) teachers.
What is the probability that a randomly selected
male teacher is shorter than a randomly selected
female teacher?
Ways to Assess Normality
Normal Probability Plot
4) What do you notice about the shape?
Normal distributions can be used to estimate
probabilities for binomial distributions when:
1) the probability of success is close to .5
or
Are these approximately normally distributed?
50
48
54
47
51
52
46
53
52
51
48
48
54
55
57
45
53
50
47
49
50
56
53
52
2) n is sufficiently large
Rule: if n is large enough,
then np > 10 & n(1 –p) > 10
Since a continuous distribution is used to estimate
the probabilities of a discrete distribution, a
continuity correction is used to make the discrete
values similar to continuous values. (  0.5 to the
discrete values)
5) Use a normal distribution with the binomial mean
and standard deviation above to estimate the
probability that between 15 & 30 preemies,
inclusive, are born in the 250 randomly selected
babies.
Normal Approximation to the Binomial
Premature babies are those born more than 3 weeks
early. Newsweek (May 16, 1988) reported that 10%
of the live births in the U.S. are premature.
Suppose that 250 live births are randomly selected
and that the number X of the “preemies” is
determined. What is the probability that there are
between 15 and 30 preemies, inclusive? (POD, p.
422)
1) Find this probability using the binomial
distribution.
2) What is the mean and standard deviation of the
above distribution?
3) If we were to graph a histogram for the above
binomial distribution, what shape do you think it will
have?
Binomial
written as
correction)
P(15 < X < 30)

Normal (w/continuity
P(14.5 < X < 30.5) =
6) How does the answer in question 6 compare to
the answer in question 1?
Homework:
A) What is the probability that less than 20
preemies are born out of the 250 babies?
B) What is the probability that at least 30 preemies
are born out of the 250 babies?
C) What is the probability that less than 35
preemies but more than 20 preemies are born out
of the 250 babies?
Homework:
A) What is the probability that less than 20 preemies are born out of the
250 babies?
P(X < 20)
P(X < 19.5) = .1231
B) What is the probability that at least 30 preemies are born out of the
250 babies?
P(X > 30)
P(X > 29.5) = .1714
C) What is the probability that less than 35 preemies but more than 20
preemies are born out of the 250 babies?
P(20 < X < 35)
P(20.5 < X < 34.5) = .8060