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Lecture slides on continuous random variables and probability distributions, including the uniform and normal distributions, calculating probabilities, assessing normality, and approximating the binomial distribution using the normal distribution. It also covers the concept of continuous probability density functions and their relationship to probability.
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Continuous Random Variables
Distribution
Continuous Random Variable Probability
P c x d f x dx c
d ( ) (^) ( )
f(x)
x c d
Probability is Area Under Curve!
Continuous Probability Distribution
Uniform Normal Exponential Other
Example on Uniform Distribution
Suppose You’re production manager of a soft
drink bottling company.
You believe that when a machine is set to
dispense 12 oz., it really dispenses 11.5 to 12.
oz. inclusive.
Suppose the amount dispensed has a uniform
distribution.
What is the probability that less than 11.8 oz. is
dispensed?
Uniform Distribution Solution
1
1 1
d c
% How to generate uniform dis. % The domain is generated x = -1:0.1:11; % now the pdf for x values pdf = unifpdf(x, 0, 10); cdf = unifcdf(x, 0, 10); subplot(1,2,1),plot(x,pdf) title('pdf') xlabel('X'), ylabel('f(x)') axis([-1 11 0 0.2]) axis square subplot(1,2,2),plot(x,cdf) title('cdf') xlabel('X'), ylabel('f(x)') axis([-1 11 0 1.1]) axis square
MATLAB Program to Generate Uniform pdf & cdf
Averages of Continuous Random Variables
The average value of a continuous random variable in an interval [a, b] is
Where, f(x) is the probability density function (pdf) for x. The normalization condition is
The average and variance value of any function of g(x) with this pdf are
b a
b a x f x dx
E x xdF x
( )
( ) ( )
var{ g } E ( g^2 ) E ( g )^2
(^ ) ^1 ()
f x dx F
E g g x f x dx
b
( ) ( ) ( )
Importance of Normal Distribution
Phenomena
Distributions Example: Binomial
Normal Distribution
Symmetrical
Mode Are Equal
Has Infinite Range
Mean, Median, Mode
x
Pdf and cdf
Cumulative Probability func (cdf) is
Probability Density Function (pdf) is
F x x dx
x
(^) 0
2
2 2
exp ( ) 2
( )^1
2
Effect of Varying Parameters ( & )
x
f(x)
A C
B
Normal distributions differ by mean & standard deviation.
x
f(x)
A C
B
f(X)
Normal distributions differ by mean & standard deviation.
Each distribution would require its own table.
That’s an infinite number!