Transcript uniform distribution

CHAPTER 7
CONTINUOUS PROBABILITY DISTRIBUTIONS
Outline
• Uniform distribution
• Normal distribution
• Exponential distribution
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UNIFORM DISTRIBUTION
• The probability density function
– f(x) and the area
• Given x, find probability
• Given probability, find x
2
UNIFORM DISTRIBUTION
THE PROBABILITY DENSITY FUNCTION
• If a random variable X is uniformly distributed over an
interval a  x  b, then its probability density function is
given by
1
f x  
a xb
ba
d c
Pc  x  d  
ba
f(x)
• The probability P(cxd)
is obtained as follows:
1
b-a
Area = 1.00
• The probability P(x=c)=0
a
3b
UNIFORM DISTRIBUTION
GIVEN x, FIND PROBABILITY
f(x)
Example 1: A retailer has observed that the monthly demand
of an item is uniformly distributed between 500 to 800
units. What is the probability that the demand of the item in
the next month will be between 600 and 700 units?
P600  x  700  ?
1
300
Area
=?
500
600
700
4
800
UNIFORM DISTRIBUTION
GIVEN x, FIND PROBABILITY
f(x)
Example 2: A retailer has observed that the monthly demand
of an item is uniformly distributed between 500 to 800
units. What is the probability that the demand of the item in
the next month will not exceed 700 units?
Px  700  ?
1
300
Area = ?
500
700
5
800
UNIFORM DISTRIBUTION
GIVEN x, FIND PROBABILITY
f(x)
Example 3: A retailer has observed that the monthly demand
of an item is uniformly distributed between 500 to 800
units. What is the probability that the demand of the item in
the next month will exceed 600 units?
Px  600  ?
1
300
Area = ?
500
600
6
800
UNIFORM DISTRIBUTION
GIVEN PROBABILITY, FIND x
f(x)
Example 4: A retailer has observed that the monthly demand
of an item is uniformly distributed between 500 to 800
units. If the retailer wants to limit the probability of stock
out to 0.10, how many units should be ordered for the next
month? Assume that there is no units in the inventory.


P x  x*  0.10
Area=.10
1
300
500
x*=? 7800
UNIFORM DISTRIBUTION
THE MEAN AND VARIANCE
ab
• The mean,  is obtained as follows:   E  X  
2
2


b

a
• The variance, 2 is obtained as follows:   V  X  
2
• An interpretation: Generate 500 random numbers between
0 and 1 using Excel. Type =RAND() in cell A1 and copy and
paste it to cells A2:A500. Check if you get the same mean
using a=0 and b=1 in the E(X) formula and Excel function
AVERAGE(A1:A500). Similarly, check if you get the same
variance using a=0 and b=1 in the V(X) formula and Excel
function VAR(). VAR() uses the sample variance formula.
The population variance is VAR(A1:A500)*499/500.
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READING AND EXERCISES
• Reading: pp. 257-262
• Exercises: 7.2,7.4
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