4-1 Continuous Random Variables
4-2 Probability Distributions and
Probability Density Functions
Figure 4-2 Probability determined from the area
under f(x).
4-2 Probability Distributions and
Probability Density Functions
Definition
4-2 Probability Distributions and
Probability Density Functions
Figure 4-3 Histogram approximates a probability
density function.
4-2 Probability Distributions and
Probability Density Functions
4-2 Probability Distributions and
Probability Density Functions
Example 4.1
Let the continuous random variable X denote the current
measured in a thin copper wire in milliamperes. Assume that
the range of X is [0,20 mA], and assume that the probability
density function of X is f(x) = 0.05 for 0 x 20.
What is the probability that a current measurement is less than 10
milliamperes?
And the probability that a current measurement is more than 5
and less than 20?
4-2 Probability Distributions and
Probability Density Functions
Example 4-2
4-2 Probability Distributions and
Probability Density Functions
Figure 4-5 Probability density function for Example 4-2.
4-2 Probability Distributions and
Probability Density Functions
Example 4-2 (continued)
4-3 Cumulative Distribution
Functions
Definition
4-3 Cumulative Distribution
Functions
Example 4.3
For the copper current measurement in Ex 4.1, the cumulative
distribution function of the random variable X consists of three
expressions. If x<0, f(x) = 0. Therefore,
0 , x0
F x 0.05 x , 0 x 20
1 , 20 x
4-3 Cumulative Distribution
Functions
Example 4-4
4-3 Cumulative Distribution
Functions
Figure 4-7 Cumulative distribution function for Example
4-4.
4-3 Cumulative Distribution
Functions
Example 4.5
The time until a chemical reaction (in milliseconds) is
approximated by the cumulative distribute function:
0 ,x 0
F x 0.01 x
1 e ,0 x
Find the probability density function of X. What proportion
of reactions is complete within 200 milliseconds.
4-4 Mean and Variance of a
Continuous Random Variable
Definition
4-4 Mean and Variance of a
Continuous Random Variable
Example 4-6
4-4 Mean and Variance of a
Continuous Random Variable
Example 4-8
4-5 Continuous Uniform Random
Variable
Definition
4-5 Continuous Uniform Random
Variable
Figure 4-8 Continuous uniform probability density
function.
4-5 Continuous Uniform Random
Variable
Mean and Variance
4-5 Continuous Uniform Random
Variable
Example 4-9
4-5 Continuous Uniform Random
Variable
Figure 4-9 Probability for Example 4-9.
4-5 Continuous Uniform Random
Variable
4-6 Normal Distribution
Definition
4-6 Normal Distribution
Figure 4-10 Normal probability density functions for
selected values of the parameters and 2.
4-6 Normal Distribution
4-6 Normal Distribution
4-6 Normal Distribution
Some useful results concerning the normal distribution
4-6 Normal Distribution
4-6 Normal Distribution
Definition : Standard Normal
4-6 Normal Distribution
Example 4-11
Figure 4-13 Standard normal probability density function.
4-6 Normal Distribution
4-6 Normal Distribution
4-6 Normal Distribution
4-6 Normal Distribution
4-6 Normal Distribution
Standardizing
4-6 Normal Distribution
Example 4-13
4-6 Normal Distribution
Figure 4-15 Standardizing a normal random variable.
4-6 Normal Distribution
To Calculate Probability
4-6 Normal Distribution
Example 4-14
4-6 Normal Distribution
Example 4-14 (continued)
4-6 Normal Distribution
Example 4-14 (continued)
Figure 4-16 Determining the value of x to meet a
specified probability.
4-6 Normal Distribution
4-6 Normal Distribution
4-6 Normal Distribution
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.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-19 Normal
approximation to the
binomial.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-17
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Binomial Distribution
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-18
X 0.5 160 151 0.5 160
P X 150 P
160 1 105
160 1 10
5
P Z 0.75 P Z 0.75 0.7733
4-7 Normal Approximation to the
Binomial and Poisson Distributions
4-7 Normal Approximation to the
Binomial and Poisson Distributions
4-7 Normal Approximation to the
Binomial and Poisson Distributions
4-7 Normal Approximation to the
Binomial and Poisson Distributions
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-21 Conditions for approximating hypergeometric
and binomial probabilities.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Poisson Distribution
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-20
949
e 1000 .1000 x
P X 950
x 0 x!
4-8 Exponential Distribution
Definition
4-8 Exponential Distribution
Mean and Variance
4-8 Exponential Distribution
Example 4-21
4-8 Exponential Distribution
Figure 4-23 Probability for the exponential distribution in
Example 4-21.
4-8 Exponential Distribution
Example 4-21 (continued)
4-8 Exponential Distribution
Example 4-21 (continued)
4-8 Exponential Distribution
Example 4-21 (continued)
