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Continuous Probability Density Fucntions
A continuous probability density functions (PDF) is a function f(x) such that f ( x)  0 on a given
b
interval [ a, b] and
 f ( x)dx  1 .
a
The Properties:
1) The mode is the value of x at the maximum value of f(x) on [ a, b] .
m
2) The median m is the solution for m of equation
1
 f ( x)dx  2
a
b
3) The mean is defines as  xf ( x)dx  
a
b
4) The variance is defined as  x 2 f ( x)dx   2  var( x)
a
0.2 x( x  b) 0  x  b
Ex) f ( x)  
is a probability density function.
0
elsewhere

a) Find b.
b) Find the mean
c) Find the variance and the standard deviation.
kx 2 ( x  6) 0  x  5
Practice) f ( x)  
is a probability density function.
0
elsewhere

a) Find k.
b) Find the mode
c) Find the mean
d) Find the variance and the standard deviation.
The Normal Distribution (Distribution for a continuous random variable)
Manu naturally occurring phenomena have a distribution that is normal or close to normal (a
Bell-shaped distribution).
The Standard Normal Curve
f ( z) 
1 ( z2 2 )
x
(for   z   ) and where z 
(Standardized variable)
e

2
Using the standard Normal (Use Graphing Calculator)

p( z  2.)  0.9773
This means :
2
1)

f ( z )dz  0.9773

2) The percentage of values of z less than 2 is 97.73%.
3) The probability (randomly chosen value of the standard normal variable less than
2) is 0.9772 .
Ex) Use “Area under the standard normal curve” table
a.
Find p( z  1.7)
b. Find p ( z  0.88)
b. Find p ( z  1.53)
d. Find p(1.7  x  2.5)
Answers:
a.
.0446
b. .1894
c. .9370
d. .0384
Finding Probabilities using the normal distribution function;
x
1 (
)
1
e 2  dx
a  2
b
p ( a  x  b)  
Ex) X is a normal random variable with mean 80 and variance 16. Find p ( x  78) .
Solution: Change the x value of 78 to Z value ( z 
x  80
) and then use the table to find
4
7880
1 (
)
1
2
4
p( z  .5)  .3085 . Or Evaluate 
e
dx
 4 2
78
Answer: .3085
The Standard Normal Distribution:
Notation: X
Ex) Given X N (32, 25) ;
a. Find P( x  18)
b. Find P(18  x  40)
N ( ,  2 )
Answers: a. 0.9974 b. 0.9426
Ex) A production line produces bags of sugar with a mean weight of 1.01 kg and standard deviation of
0.02 Kg.
a. Find the proportion of the bags that weight less than 1.03 Kg.
b. Find the proportion of the bags that weigh more than 1.02 kg.
c. Find the percent of the bags that weigh between 1.00 kg and 1.05 kg.
Answers: a. 0.8413
b. 0.3085
c. .6687
Inverse to find Quartiles;
Ex) Find the values of a in each of these statements that refer to the standard normal
variables, Z. (Use “Inverse Normal probabilities table)
a.
b.
c.
p( z  k )  0.5478
p( z  k )  0.6
p( z  k )  0.05
Answers: a. 0.1201
b. -.2533
c. -1.645
Ex) If X N (100, 25) , find the value of k such that
a. p( x  k )  0.9
b. p( x  k )  0.2
Answers: a. 106
b. 95.8
Ex) The life time of a particular make of television tube is normally distributed with a
mean of 8 years, and a standard deviation of n years. The chances that the tube will not
last 5 years are 0.05. What is the value of the standard deviation, n?
Ex) The weight of a population of men is found to be normally distributed with mean 69.5
kg. 13 % of the men weight at least 72.1 Kg. Find the standard deviation of their weight?