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Calculus 221: Normal Random Variables and the Normal Distribution - Prof. Timothy John Pil, Study notes of Calculus

University of MarylandCalculus
Prof. Timothy John Pilachowski

Notes on the normal random variables and the normal distribution, including the normal density function, expected value, standard deviation, and examples of finding probabilities using normal distributions. It also discusses the relationship between the mean and standard deviation and the concept of a standard normal distribution.

Typology: Study notes

Pre 2010

Uploaded on 07/30/2009

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Calculus 221, section 12.4b Normal Random Variables
notes prepared by Tim Pilachowski
Another often-useful probability density function is the normal density function,
which graphs as the familiar bell-shaped curve. The generic format is
()
2
2
1
2
1⎟
⎠
⎞
⎜
⎝
⎛−
−
=
σ
µ
πσ
x
exf , where E(X) =
µ
, Var(X) = σ2, and standard
deviation
()
σ
== XVar .
The graph of a normal curve is symmetric with respect to the line
µ
=
x
, and h
points of inflection at
as
σ
µ
−=
x
and
σ
µ
+
=
x
.
Example A: Find the expected value and standard deviation of the normal random variable X with probability
density function
()
2
3
5
2
1
23
1⎟
⎠
⎞
⎜
⎝
⎛−
−
=
x
exf
π
. answers: 5; 3
0.5
–
10
0
10
Example B: Find the expected value and standard deviation of the normal random variable X with probability
density function
()
()
2
2
2
1
2
1−−
=x
exf
π
. answers: 2; 1
0.5
–
10
0
10
Example C: Find the expected value and standard deviation of the normal random variable X with probability
density function
()
2
72
1
26
1x
exf −
=
π
. answers: 0; 6
0.5
–
10
0
10
pf3
pf4

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Calculus 221, section 12.4b Normal Random Variables notes prepared by Tim Pilachowski Another often-useful probability density function is the normal density function,which graphs as the familiar bell-shaped curve. The generic format isdeviation The graph of a normal curve is symmetric with respect to the linepoints of inflection at Example A: Find the expected value and standard deviation of the normal random variable f ( ) x =σ = 1 2 πVar e −( 1 2 X ⎜⎝⎛) xx σ−= =μ σ⎟⎠⎞ μ 2. −, where E( σ and x X =) = μ + μ σ, Var(. X ) = σ 2 , and standard x = μ, and h as X with probability

density function Example B: Find the expected value and standard deviation of the normal random variabledensity function ff ( )( ) xx == 3211 π 2 π e − e − 2121 ( x ⎜⎝⎛−^^ x 23 −)^52 ⎟⎠⎞. 2. answers answers : 2; 1: 5; 3^ 0.5 0.5 – 100 X with probability 10

Example C: Find the expected value and standard deviation of the normal random variabledensity function f ( ) x = 6 12 π e − 721 x 2. answers : 0; 6 0.5^ – –^100100 X with probability^1010

If we consider the special case where E(1, we get what is called the standard normal distribution,with its graph called the standard normal curve.Differences in the means result in shifts left and right. A smaller standarddeviation will result in a taller, more narrow “bell”. Each curve is symmetricabout the mean. Note that in all three cases, probabilities beyondso small as to usually be considered insignificant. X ) = μ = 0 and standard deviation = f ( ) x = 21 π e − 21 x σ 2 =,^ μ^ ±^3 σ become

There is no method for integrating to find area under the curve (and thusprobabilities) of this standard normal probability density function, but acalculation using a Taylor polynomial (covered in Math 221) has been done toconstruct tables of values such as the one found in Appendix E of your text. For a standardized random variable In an Excel spreadsheet, the function =NORMSDIST(z) gives area under thestandard normal curve = probability for the interval from – Examples D: Given a standard normal random variablec) Pr(–1.23 A ( − z ) = A ( )≤ z Z = ≤Pr 0) d) Pr(–2.14 ( 0 ≤ Z ≤ z ) = area under the curve from the mean to ≤ Z Z ≤, 1.23). answers : 0.3907; 0.8907; 0.3907; 0.8745 Z , find a) Pr(0∞ to z. ≤ zZ. ≤ 1 .23) b) Pr( Z ≤ 1.23)

12.4b Appendix E

  • 0.00 0.01 0.02 0.03 0. z Appendix ETable 1Areas underthe StandardNormal Curve A ( z )
  • 0.05 0.06 0.07 0.08 0.
    • 0.0 0.0000 0.0040 0.0080 0.0120 0.
    • 0.0199 0.0239 0.0279 0.0319 0.
      • 0.1 0.0398 0.0438 0.0478 0.0517 0.
      • 0.0596 0.0636 0.0675 0.0714 0.
        • 0.2 0.0793 0.0832 0.0871 0.0910 0.
        • 0.0987 0.1026 0.1064 0.1103 0.
          • 0.3 0.1179 0.1217 0.1255 0.1293 0.
          • 0.1368 0.1406 0.1443 0.1480 0.
            • 0.1554 0.1591 0.1628 0.1664 0.1700 0.
            • 0.1736 0.1772 0.1808 0.1844 0.
              • 0.1915 0.1950 0.1985 0.2019 0.2054 0.
              • 0.2088 0.2123 0.2157 0.2190 0.
                • 0.6 0.2257 0.2291 0.2324 0.2357 0.
                • 0.2422 0.2454 0.2486 0.2517 0.
                  • 0.7 0.2580 0.2611 0.2642 0.2673 0.
                  • 0.2734 0.2764 0.2794 0.2823 0.
                    • 0.8 0.2881 0.2910 0.2939 0.2967 0. - 0.3023 0.3051 0.3078 0.3106 0. - 0.9 0.3159 0.3186 0.3212 0.3238 0. - 0.3289 0.3315 0.3340 0.3365 0. - 1.0 0.3413 0.3438 0.3461 0.3485 0. - 0.3531 0.3554 0.3577 0.3599 0. - 1.1 0.3643 0.3665 0.3686 0.3708 0. - 0.3749 0.3770 0.3790 0.3810 0. - 1.2 0.3849 0.3869 0.3888 0.3907 0. - 0.3944 0.3962 0.3980 0.3997 0. - 0.4032 0.4049 0.4066 0.4082 0.4099 1. - 0.4115 0.4131 0.4147 0.4162 0. - 0.4192 0.4207 0.4222 0.4236 0.4251 1. - 0.4265 0.4279 0.4292 0.4306 0. - 1.5 0.4332 0.4345 0.4357 0.4370 0. - 0.4394 0.4406 0.4418 0.4429 0. - 1.6 0.4452 0.4463 0.4474 0.4484 0. - 0.4505 0.4515 0.4525 0.4535 0. - 1.7 0.4554 0.4564 0.4573 0.4582 0. - 0.4599 0.4608 0.4616 0.4625 0. - 1.8 0.4641 0.4649 0.4656 0.4664 0. - 0.4678 0.4686 0.4693 0.4699 0. - 1.9 0.4713 0.4719 0.4726 0.4732 0. - 0.4744 0.4750 0.4756 0.4761 0. - 2.0 0.4772 0.4778 0.4783 0.4788 0. - 0.4798 0.4803 0.4808 0.4812 0. - 2.1 0.4821 0.4826 0.4830 0.4834 0. - 0.4842 0.4846 0.4850 0.4854 0. - 0.4861 0.4864 0.4868 0.4871 0.4875 2. - 0.4878 0.4881 0.4884 0.4887 0. - 0.4893 0.4896 0.4898 0.4901 0.4904 2. - 0.4906 0.4909 0.4911 0.4913 0. - 2.4 0.4918 0.4920 0.4922 0.4925 0. - 0.4929 0.4931 0.4932 0.4934 0. - 2.5 0.4938 0.4940 0.4941 0.4943 0. - 0.4946 0.4948 0.4949 0.4951 0. - 2.6 0.4953 0.4955 0.4956 0.4957 0. - 0.4960 0.4961 0.4962 0.4963 0. - 2.7 0.4965 0.4966 0.4967 0.4968 0. - 0.4970 0.4971 0.4972 0.4973 0. - 2.8 0.4974 0.4975 0.4976 0.4977 0. - 0.4978 0.4979 0.4979 0.4980 0. - 2.9 0.4981 0.4982 0.4982 0.4983 0. - 0.4984 0.4985 0.4985 0.4986 0. - 0.4987 0.4987 0.4987 0.4988 0.4988 3. - 0.4989 0.4989 0.4989 0.4990 0. - 0.4990 0.4991 0.4991 0.4991 0.4992 3. - 0.4992 0.4992 0.4992 0.4993 0. - 0.4993 0.4993 0.4994 0.4994 0.4994 3. - 0.4994 0.4994 0.4995 0.4995 0. - 3.3 0.4995 0.4995 0.4995 0.4996 0. - 0.4996 0.4996 0.4996 0.4996 0. - 3.4 0.4997 0.4997 0.4997 0.4997 0. - 0.4997 0.4997 0.4997 0.4997 0. - 3.5 0.4998 0.4998 0.4998 0.4998 0. - 0.4998 0.4998 0.4998 0.4998 0.