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Binomial Random Variable Approximations, Conditional Probability Density Functions and Stirling’s Formula.

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Presentation on theme: "Binomial Random Variable Approximations, Conditional Probability Density Functions and Stirling’s Formula."— Presentation transcript:

1 Binomial Random Variable Approximations, Conditional Probability Density Functions and Stirling’s Formula

2  Let X represent a Binomial r.v,Then from  for large n. In this context, two approximations are extremely useful.

3 The Normal Approximation (Demoivre-Laplace Theorem)  Suppose with p held fixed. Then for k in the neighborhood of np, we can approximate  And we have:  where

4  As we know,  If and are within with approximation:  where  We can express this formula in terms of the normalized integral  that has been tabulated extensively.

5

6  A fair coin is tossed 5,000 times.  Find the probability that the number of heads is between 2,475 to 2,525.  We need  Since n is large we can use the normal approximation.  so that and  and  So the approximation is valid for and Example Solution

7  Here,  Using the table, Example - continued

8 The Poisson Approximation  For large n, the Gaussian approximation of a binomial r.v is valid only if p is fixed, i.e., only if and  What if is small, or if it does not increase with n ? -for example, as such that is a fixed number.

9 The Poisson Approximation  Consider random arrivals such as telephone calls over a line.  n : total number of calls in the interval  as we have  Suppose  Δ : a small interval of duration

10 The Poisson Approximation  p : probability of a single call (during 0 to T) occurring in Δ:  as  Normal approximation is invalid here.  Suppose the interval Δ in the figure: -(H) “success” : A call inside Δ, -(T ) “failure” : A call outside Δ  : probability of obtaining k calls (in any order) in an interval of duration Δ,

11 The Poisson Approximation  Thus, the Poisson p.m.f

12  Suppose -two million lottery tickets are issued -with 100 winning tickets among them.  a) If a person purchases 100 tickets, what is the probability of winning? Example: Winning a Lottery Solution The probability of buying a winning ticket

13  X : number of winning tickets  n : number of purchased tickets,  P : an approximate Poisson distribution with parameter  So, The Probability of winning is: Winning a Lottery - continued

14  b) How many tickets should one buy to be 95% confident of having a winning ticket?  we need  But or  Thus one needs to buy about 60,000 tickets to be 95% confident of having a winning ticket! Winning a Lottery - continued Solution

15  A space craft has 100,000 components  The probability of any one component being defective is  The mission will be in danger if five or more components become defective.  Find the probability of such an event.  n is large and p is small  Poisson Approximation with parameter Example: Danger in Space Mission Solution

16 Conditional Probability Density Function

17  Further,  Since for

18  Toss a coin and X ( T )=0, X ( H )=1.  Suppose  Determine  has the following form.  We need for all x.  For so that  and Example Solution (a) 1 1 (b) 1 1

19  For so that  For and Example - continued 1 1

20  Given suppose Find  We will first determine  For so that Example Solution

21  Thus  and hence Example - continued (a) (b)

22  Let B represent the event with  For a given determine and Example Solution

23  For we have and hence  For we have and hence  For we have so that  Thus, Example - continued

24 Conditional p.d.f & Bayes’ Theorem  First, we extend the conditional probability results to random variables:  We know that If is a partition of S and B is an arbitrary event, then:  By setting we obtain:

25 Conditional p.d.f & Bayes’ Theorem  Using:  We obtain:  For,

26 Conditional p.d.f & Bayes’ Theorem  Let so that in the limit as  or  we also get  or (Total Probability Theorem)

27 Bayes’ Theorem (continuous version)  using total probability theorem in  We get the desired result

28  probability of obtaining a head in a toss.  For a given coin, a-priori p can possess any value in (0,1).  : A uniform in the absence of any additional information  After tossing the coin n times, k heads are observed.  How can we update this is new information?  Let A = “ k heads in n specific tosses”.  Since these tosses result in a specific sequence,  and using Total Probability Theorem we get Example: Coin Tossing Problem Revisited Solution

29  The a-posteriori p.d.f represents the updated information given the event A,  Using  This is a beta distribution.  We can use this a-posteriori p.d.f to make further predictions.  For example, in the light of the above experiment, what can we say about the probability of a head occurring in the next ( n +1)th toss? Example - continued

30  Let B = “head occurring in the ( n +1)th toss, given that k heads have occurred in n previous tosses”.  Clearly  From Total Probability Theorem,  Using (1) in (2), we get:  Thus, if n =10, and k = 6, then  which is more realistic compare to p = 0.5. Example - continued


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