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Probability questions: Gaussian approximation and Poisson approximation

Probability:

1.You flip a coin n times.Please answer the following questions about the fraction f of the tosses that show heads.

A. Lets say n=100. Compute exactly the prob that the fraction of the coins showing heads is f=0.5
B. Say n=100 and consider the prob that f=0.5. Show that this probability can be approximated using the Gaussian function by showing that the conditions required for this approximation are met. Find the Gaussian approximation for the probability. What is the approximation error?
C.Repeat parts a and b but with n=10000 and f=0.5.
D. As n gets larger, does the probability that we get exactly n/2 heads decrease or increase? (you may assume that n is even).
E. As n gets larger, does the prob that 0.49<=f<=0.51 increase or decrease?

2.
A. A system consists of 1000 components each of which fails independently with prob 1/2000. Using Poisson approximation, determine the probability that this system works (i.e. that all components work).
B. We decide the prob of failure of the system in part a) is too high, so we decide to make n copies of the system( which operate independently). What is the minimum n such that the probability of at least one copy works is greater than or equal to 0.99?
C. Instead of building n copies of the entire system, you decide to build redundancy into each of the components.In particular, you make n copies of each component and put them together in such a way that the component works if any of its n copies work.Is this a better or worse strategy than making n copies of the entire system?

3. A geometric random variable X with parameter q (0<q<1) has the foll PMF:
P(X=w) = (1-q)q^w,, w= 0,1,2.....
A.Find E[X].
B.given tht X>=2, wht is the PMF of X?given tht X>=2, wht is the PMF of X-2?

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Probability:

1.You flip a coin n times.Please answer the following questions about the fraction f of the tosses that show heads.

A.Lets say n=100.Compute exactly the prob tht the fraction of the coins showing heads is f=0.5
B. Say n=100 and consider the prob tht f=0.5. show tht this prob can be approximated using the gaussian function by showing tht the conditions reqd for this approximation are met.Find the gaussian approximation for the prob.Wht is the approximation error?
C.Repeat parts a and b but with n=10000 and f=0.5.
D. As n gets larger,does the prob tht we get exactly n/2 heads decrease or increase?(u may assume tht n is even).
E. as n gets larger,does tht prob tht 0.49<=f<=0.51 increase or decrease?

2.
A. A system consists of 1000 components each of which fails independently with prob 1/2000. Using poisson approximation, determine the prob tht this system works(i.e. that all components work).
B. We decide the prob of failure of the system in part a) is too high, so we decide to make n copies of the system( which operate independently).wht is the minimum n such tht the prob tht at least one copy works is greater than or equal to 0.99?
C. Instead of building n copies of the entire system, u decide to build redundancy into each of the components.In particular,u make n copies of each component and put them together in such a way that the component works if any of its n copies work.Is this a better or worse strategy thn making n copies of the entire system?

3.A geometric andom variable X with parameter q (0<q<1) has the foll PMF:
P(X=w) = (1-q)q^w,, w= 0,1,2.....
A.Find E[X].
B.given tht X>=2, wht is the PMF of X?given tht X>=2, wht is the PMF of X-2?

Question 1
a. If n = 100, then f = 0.5 when the number of successes is equal to 50. In other words, calling X to the number heads that come after flipping a coin n ...

Solution Summary

Two series of questions involving calculations using Gaussian approximation and Poisson approximations to calculate certain probabilities.

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