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Ch7 Continuous Probability Distributions 連續型機率分配(P210/P198) - Coggle…
Ch7 Continuous Probability Distributions 連續型機率分配(P210/P198)
1 Intro
Usually results from
measuring
something.透過
測量
得知 (P210/P198)
Properties:
(P210/P198)
3 The probability for
a specific value
of a continuous random variable is 0.
2 Probability is
for a range of value
s.
1 A continuous random variable
has an infinite number of values within a particular range
.
Three continuous probability distributions(P210/P198)
2 The normal probability distribution （常態機率分配）
(P216/P203)
3 The exponential probability distribution （指數機率分配）
1 The uniform probability distribution (均勻機率分配)
(P211/P198)
1 The uniform probability distribution (均勻機率分配)＝Information-less distribution (沒有資訊的機率分配)
(P211/P198~199)
Application Situation:
Don't have any information
regarding the shape of a random variable’s probability distribution.(P211/P198~199)
Characteristics
: (P239/P228)
The
mean
and the
median
are
equal
.
It is completely described by its
minimum value
a and its
maximum value
b.
It is
rectangular
in shape.
How to calculate:
The Standard Deviation
(P212/P200)
Related to the interval between the minimum and maximum values.
7-3
Mean
(P212/200)
Located in the exact middle of the range between a and b.
7-2
EX: (P213/P200)
A bus arrives at the North Main Street and College Drive stop every 30 minutes between 6 a.m. and 11 p.m. during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 minutes.
Probability (P211/P199)
Areas within or under the distribution. (
Chart 7-1
)
The total area within a continuous probability distribution is equal to
1.00
.
(7-1)
2 The normal probability distribution （常態機率分配）
(P216/P203)
Characteristics
: (P216/P204)
Symmetrical
Bell shaped
and
A single peak
Asymptotic
Determined by the
mean, μ
and the
standard deviation, σ.
Chart 7-3 (P216/P204)
EX: Chart7-4, 7-5, 7-6 (P216/P204)
How to calculate:
The normal probability distribution （常態機率分配）
7-4
(P216/P204)
The symbols μ and σ refer to the mean and the standard deviation, as usual. The Greek symbol π is a constant and its value is approximately 22/7 or 3.1416. The letter e is also a constant. It is the base of the natural log system and is approximately equal to 2.718. x is the value of a continuous random variable.
Use a table
, given in Appendix B.3
Use software packages or online calculators.
The Standard Normal Probability Distribution（標準常態機率分配）
(P218/P206)
Mean = 0
Standard Deviation of 1
Variance equal to 1
A normal distribution with a mean equal to 0 and variance equal to 1.
Z Value = Z Scores = Standard Normal Values (P218/P206
)
z VALUE The signed distance between a selected value, designated x, and the mean, μ, divided by the standard deviation, σ
The z value calculates the distance between a value of the random variable, x, and the mean of the distribution, μ, in units of the standard deviation, σ.
7-5
Procedures:
(P218/P207)
3 more items...
x is the value of a normally distributed random variable. μ is the mean of the normal distribution. σ is the standard deviation of the normal distribution.
z value expresses the distance or difference between a particular value of x and the arithmetic mean in units of the standard deviation
The z distribution has
all the characteristics of any normal probability distribution
. It is bell shaped, symmetrical, and asymptotic.
Any normal probability distribution can be converted into a standard normal probability distribution
by subtracting the mean from each observation and dividing this difference by the standard deviation.
The Empirical Rule
(P231/P220, Ch3 P55/P81~84)
About 95% of the observations will lie within plus and minus 2 standard deviations of the mean.
(P232/P220)
Verify the Empirical Rule
Practically all, or 99.7% of the observations, will lie within plus and minus 3 standard deviations of the mean.
Approximately 68% of the observations will lie within plus and minus 1 standard deviation of the mean.
1 standard deviation from the mean is the same as a z value of 1.00. When we refer to the standard normal probability table (Appendix B.3), a z value of 1.00 corresponds to a probability of 0.3413. Multiply (2)(0.3413), which equals 0.6826, or approximately 68% of the observations are within plus and minus 1 standard deviation of the mean.
3 Exponential Distributions（指數分配）
(P234/P222)
Application Situations: Usually describes times between events in a sequence.
The actions occur
independently
at a constant rate per unit of time or length.
Characteristics
: (P234/P222)
Because time is never negative, an exponential random variable is
always positive
. (P234/P222)
The exponential probability distribution is
positively skewed
.(P234/P222)
The distribution is described by only one parameter, which we will identify as λ. (P234/P222)
λ is often referred to as the “rate” parameter.
Decrease λ, the shape of the distribution is “less skewed.
(P234/P222)
The graph of the exponential distribution starts at the value of λ when the random variable’s (x) value is 0. The distribution declines steadily as we move to the right with increasing values of x. (P234/P223)
Close relationship to the Poisson distribution. (P234/P223)
Declines steadily to the righ
t, and is
asymptotic (漸進的)
.(P239/P228)
How to calculate:
Variance =
(P239/P229)
Formula 7–6
describes the exponential probability distribution with λ as rate parameter.
(P235/P223)
Mean＝
(P239/P229)
Find a Probability using the exponential distribution
7-7
(P235/P223)
Finding the percentage of the observations located between two values or the percentage of the observations above or below a particular value, x. （求介於兩個值的觀測值百分比）
EX:Orders for prescriptions arrive at a pharmacy website (P235/P223~225)
“Reverse” to find the value of the observation x when the percentage above or below the observation is given. （求已知高於或低於某觀測值的百分比時，反推出此觀測值的數值是多少？）(P236/P225)
EX: Compton Computers wishes to set a minimum lifetime guarantee on its new power supply unit.(P236/P226)
The complement rule （互補規則）is applied as follows: (P236/P224)
