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Let X represent a Binomial r.v ,Then from

Binomial Random Variable Approximations, Conditional Probability Density Functions and Stirling’s Formula. Let X represent a Binomial r.v ,Then from for large n . In this context, two approximations are extremely useful. The Normal Approximation (Demoivre-Laplace Theorem).

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Let X represent a Binomial r.v ,Then from

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  1. Binomial Random Variable Approximations,Conditional Probability Density Functionsand 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. Example • 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 Solution

  6. Example - continued • Here, • Using the table,

  7. 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.

  8. 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

  9. 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 ,

  10. The Poisson Approximation • Thus, the Poisson p.m.f

  11. Example: Winning a Lottery • 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? Solution The probability of buying a winning ticket

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

  13. Winning a Lottery - continued • 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! Solution

  14. Example: Danger in Space Mission • 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 Solution

  15. Conditional Probability Density Function

  16. Conditional Probability Density Function • Further, • Since for

  17. 1 1 1 1 (b) (a) Example • 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 Solution

  18. 1 1 Example - continued • For so that • For and

  19. Example • Given suppose Find • We will first determine • For so that • For so that Solution

  20. (b) (a) Example - continued • Thus • and hence

  21. Example • Let B represent the event with • For a given determine and Solution

  22. Example - continued • For we have and hence • For we have and hence • For we have so that • Thus,

  23. 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:

  24. Conditional p.d.f & Bayes’ Theorem • Using: • We obtain: • For ,

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

  26. Bayes’ Theorem (continuous version) • using total probability theorem in • We get the desired result

  27. Example: Coin Tossing Problem Revisited • 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 Solution

  28. Example - continued • 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?

  29. Example - continued • 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.

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