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Continuous Probability
Distribution
* Uniform Distribution
* Normal Distribution
Definition
4-5 Continuous Uniform Distribution
- Discrete uniform distribution, f(x) = 1/n
Figure 4-8 Continuous uniform probability density
function.
4-5 Continuous Uniform Distribution
Discrete uni. dis
Mean and Variance
4-5 Continuous Uniform Distribution
Example
4-5 Continuous Uniform Distribution
To know that it is unifm. dist
f(x)=1/ (b-a)
=1/ (20-0) =0.05
Figure 4-9 Probability for Example 4-9.
4-5 Continuous Uniform Distribution
4-5 Continuous Uniform Distribution
Text Book Problem
• 441
a. F(X) = (x-0.9)/0.15
b. P(X>1.02) = 0.2
c. Here you have to identify the value of x such
that the function will equate to 0.9 or 90%
i.e., P(X>x)=0.9
x=0.915
d. E(X)=0.976; V(X)=0.00188
Text Book Problem
• 446 (Please note that there is a mistake in
the problem. Change 746 with 374 and 752
with 380)
a. E(X) = 377; s(X)=1.732
b. P(X<375) = 0.1667
c. P(X>x)=0.95; x=374.3
d. Mean extra cost = 0.004/ container
(Hint: mean extra cost = cost* mean extra
volume)
Normal
Distribution
• ‘Bell Shaped’
• Symmetrical
• Mean, Median and Mode
are Equal
Location is determined by the
mean, μ
Spread is determined by the
standard deviation, σ
The random variable has an
infinite theoretical range:
+  to  
X
f(X)
μ
σ
4-6 Normal Distribution
4-6 Normal Distribution
Definition
4-6 Normal Distribution
Figure 4-10 Normal probability density functions for
selected values of the parameters  and s2.
f(X)
X
μ μ+1σ
μ-1σ
σ
σ
68.26%
x
μ
2σ 2σ
x
μ
3σ 3σ
95.44% 99.73%
4-6 Normal Distribution
Some useful results concerning the normal distribution
4-6 Normal Distribution
Definition : Standard Normal
z = (x-)/s
4-6 Normal Distribution
Example 4-11
Figure 4-13 Standard normal probability density function.
Normal distribution table is
in Page 718 and Page 719
P(Z<=1.5)=0.93319 P(Z<=1.53)=0.93699
P(Z<=0.54 =0.7054)
P(Z<=1.0 =0.8413)
P(Z<=-2.16 =0.015)
P(Z<=-0.91 =0.1814)
Text-book problem
4.49
a. P(Z<1.32) =0.9066
c. P(Z>1.45) = 1-P(Z<1.45) (Because the
table gives the lower value of Z i.e., P(Z<z)
= 0.0735
d. P(Z>-2.15) = 1- 0.0158=0.9842
e. P(-2.34<Z<1.76) ---- see in the next slide
1.45
1 = 100%
Non-shaded area
= 1-shaded area
f(X)
X
1.76
-2.34
a = P(Z<1.76)
P(-2.34<Z<1.76)
b = P(Z<-2.34)
P(-2.34<Z<1.76) = P(Z<1.76)-P(Z<-2.34)
= 0.9511
Thanks a lot for listening.
Remaining portion we will
discuss in the next class

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ch04 - part 2 (Cont unifm dist).ppt explained