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# Cumulative distribution

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### Cumulative distribution

1. 1. 1 Cumulative distribution function & Expectation Shashwat Shriparv dwivedishashwat@gmail.com InfinitySoft
2. 2. 2  In modeling real-world phenomena there are few situations where the actions of the entities within the system under study can be completely predicted in advance.  The world the model builder sees is probabilistic rather than deterministic
3. 3. 3  An appropriate model can be developed by sampling the phenomenon of interest.  Then, through educated guesses the model builder would select a known distribution form, make an estimate of the parameter of this distribution, and then test to see how good a fit has been obtained.
4. 4. 4 Review of terminology & concepts  Discrete random variables.  Continuous random variables.  Cumulative distribution function  Expectation
5. 5. 5 Cumulative Distribution Function
6. 6. 6  The cumulative distribution function (cdf), denoted by F(x), measures the probability that the random variable X assumes a value less than or equal to x, that is, F(x)=P(X<=x).
7. 7. 7 Properties of the cdf  F is a non-decreasing function, If a<b, then F(a)<=F(b).  Lim x->-∞ F(x)=0.  Lim x->∞ F(x)=1.
8. 8. 8 Expectation
9. 9. 9  An important concept in probability theory is that of the expectation of a random variable.  if x is a random variable, the expected value of X, is denoted by E(X).
10. 10. 10  The expected value E(X) of a random variable X is also referred to as the mean, or the first moment of X.  The variance of a random variable, X, is denoted by V(X) of var(X), is defined as V(X)=E[(X-E[X])²]  And also V(X)=E(X²)-[E(X)]²
11. 11. 11  The mean E(X) is a measure of the central tendency of a random variable.  The variance of X measures the expected value of the squared difference between the random variable and its mean.  Thus the variance, V(X), is a measure of the spread or variation of the possible values of X around the mean E(X).
12. 12. 12 Shashwat Shriparv dwivedishashwat@gmail.com InfinitySoft