4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate...
Transcript of 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate...
![Page 1: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/1.jpg)
4-1 Continuous Random Variables
![Page 2: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/2.jpg)
rv Discrete rv Continuous rv
Range of rv Finite or countably finite An interval of real
numbers
Example Toss a coin 20 times Measure the lifetime of
an electronic component
rv : X Number of heads Lifetime in hours
Range of X {0,1,2,…..20} {x: 0≤x<infinity}
pmf pdf
![Page 3: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/3.jpg)
4-2 Probability Distributions and
Probability Density Functions
Figure 4-1 Density function of a loading on a long,
thin beam.
![Page 4: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/4.jpg)
4-2 Probability Distributions and
Probability Density Functions
Figure 4-2 Probability determined from the area
under f(x).
![Page 5: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/5.jpg)
4-2 Probability Distributions and
Probability Density Functions
Definition
![Page 6: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/6.jpg)
Review pmf
![Page 7: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/7.jpg)
4-2 Probability Distributions and
Probability Density Functions
Figure 4-3 Histogram approximates a probability
density function.
![Page 8: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/8.jpg)
4-2 Probability Distributions and
Probability Density Functions
![Page 9: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/9.jpg)
4-2 Probability Distributions and
Probability Density Functions
Example 4-2
![Page 10: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/10.jpg)
4-2 Probability Distributions and
Probability Density Functions
Figure 4-5 Probability density function for Example 4-2.
![Page 11: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/11.jpg)
4-2 Probability Distributions and
Probability Density Functions
Example 4-2 (continued)
![Page 12: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/12.jpg)
4-3 Cumulative Distribution Functions
Definition
![Page 13: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/13.jpg)
4-3 Cumulative Distribution Functions
Example 4-4
![Page 14: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/14.jpg)
Figure 4-7 Cumulative distribution function for Example
4-4.
![Page 15: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/15.jpg)
Practice: The pdf for a r.v X is given by
f(x) = c x^2, 0≤x≤2
(a) Find c
(b)Find p(X≤3)
(c) Find CDF of f(x)
(d)Use CDF to find pdf of X
![Page 16: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/16.jpg)
4-4 Mean and Variance of a Continuous
Random Variable
Definition
![Page 17: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/17.jpg)
4-4 Mean and Variance of a Continuous
Random Variable
Example 4-6
![Page 18: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/18.jpg)
4-4 Mean and Variance of a Continuous
Random Variable
Expected Value of a Function of a Continuous
Random Variable
![Page 19: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/19.jpg)
4-4 Mean and Variance of a Continuous
Random Variable
Example 4-8
![Page 20: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/20.jpg)
4-5 Continuous Uniform Random
Variable
Definition
![Page 21: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/21.jpg)
4-5 Continuous Uniform Random
Variable
Figure 4-8 Continuous uniform probability density
function.
![Page 22: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/22.jpg)
4-5 Continuous Uniform Random
Variable
Mean and Variance
![Page 23: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/23.jpg)
4-5 Continuous Uniform Random
Variable
Example 4-9
![Page 24: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/24.jpg)
4-5 Continuous Uniform Random
Variable
Figure 4-9 Probability for Example 4-9.
![Page 25: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/25.jpg)
4-5 Continuous Uniform Random
Variable
![Page 26: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/26.jpg)
4-6 Normal Distribution
Definition
![Page 27: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/27.jpg)
4-6 Normal Distribution
Figure 4-10 Normal probability density functions for
selected values of the parameters and 2.
![Page 28: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/28.jpg)
4-6 Normal Distribution
Some useful results concerning the normal distribution
![Page 29: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/29.jpg)
4-6 Normal Distribution
Definition : Standard Normal
![Page 30: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/30.jpg)
4-6 Normal Distribution
Example 4-11
Figure 4-13 Standard normal probability density function.
![Page 31: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/31.jpg)
4-6 Normal Distribution
![Page 32: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/32.jpg)
Example: Let Z be a standard normal rv
N(0,1), determine the following
(1)p(Z ≤1.26)
(2)p(Z>1.26)
(3)p(Z>-1.26)
(4)P(-1.26<Z<1.26)
(5)Find value z such that p(Z>z) =.05
(6)Find z such that p(-z<Z<z)=.95
![Page 33: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/33.jpg)
4-6 Normal Distribution
Example 4-13
![Page 34: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/34.jpg)
4-6 Normal Distribution
Figure 4-15 Standardizing a normal random variable.
![Page 35: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/35.jpg)
4-6 Normal Distribution
To Calculate Probability
![Page 36: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/36.jpg)
4-6 Normal Distribution
Example 4-14
![Page 37: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/37.jpg)
4-6 Normal Distribution
Example 4-14 (continued)
![Page 38: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/38.jpg)
4-6 Normal Distribution
Example 4-14 (continued)
Figure 4-16 Determining the value of x to meet a
specified probability.
![Page 39: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/39.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
• Under certain conditions, the normal
distribution can be used to approximate the
binomial distribution and the Poisson
distribution.
![Page 40: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/40.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-19 Normal
approximation to the
binomial.
![Page 41: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/41.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-17
![Page 42: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/42.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Binomial Distribution
![Page 43: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/43.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-18
![Page 44: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/44.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-21 Conditions for approximating hypergeometric
and binomial probabilities.
![Page 45: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/45.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Poisson Distribution
![Page 46: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/46.jpg)
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-20
![Page 47: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/47.jpg)
4-8 Exponential Distribution
Definition
![Page 48: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/48.jpg)
4-8 Exponential Distribution
Mean and Variance
![Page 49: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/49.jpg)
4-8 Exponential Distribution
Example 4-21
![Page 50: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/50.jpg)
4-8 Exponential Distribution
Figure 4-23 Probability for the exponential distribution in
Example 4-21.
![Page 51: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/51.jpg)
4-8 Exponential Distribution
Example 4-21 (continued)
![Page 52: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/52.jpg)
4-8 Exponential Distribution
Example 4-21 (continued)
![Page 53: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/53.jpg)
4-8 Exponential Distribution
![Page 54: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/54.jpg)
Practice: Suppose that calls to a radio
line follow a poisson process with an
average of 1/15 calls/minute
(1)What is the mean time between calls
(2)Let x=time between two calls, what is the
probability that there are no calls within 30
minutes
(3)Probability at least one call in a 10-minute
interval
(4)What is the probability that the first call
occurs within 10 to 15 minutes?
![Page 55: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/55.jpg)
4-8 Exponential Distribution
Our starting point for observing the system does not matter.
•An even more interesting property of an exponential random
variable is the lack of memory property.
In Example 4-21, suppose that there are no log-ons
from 12:00 to 12:15; the probability that there are no
log-ons from 12:15 to 12:21 is still 0.082. Because we
have already been waiting for 15 minutes, we feel that
we are “due.” That is, the probability of a log-on in the
next 6 minutes should be greater than 0.082. However,
for an exponential distribution this is not true.
![Page 56: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/56.jpg)
4-8 Exponential Distribution
Example 4-22
![Page 57: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/57.jpg)
4-8 Exponential Distribution
Example 4-22 (continued)
![Page 58: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/58.jpg)
4-8 Exponential Distribution
Example 4-22 (continued)
![Page 59: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/59.jpg)
4-8 Exponential Distribution
Lack of Memory Property
![Page 60: 4-1 Continuous Random Variablesluyan/stat34519/ch04.pdf · 4-6 Normal Distribution To Calculate Probability. 4-6 Normal Distribution Example 4-14. 4-6 Normal Distribution Example](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7f7194b8270a5833351193/html5/thumbnails/60.jpg)
4-8 Exponential Distribution
Figure 4-24 Lack of memory property of an Exponential
distribution.