The binomial problem must be “large enough” that it behaves like something close to a normal curve. In terms of the picture that we were discussing before, what we are doing, essentially, is to take the area under the normal PDF that extends from 18.5 to 19.5 and declare that this area corresponds to the discrete event that our binomial random variable takes a value of 19. Normal Approximation to the Binomial distribution. Unformatted text preview: Normal approximation to binomial In this lecture we discuss : • 0 Using normal distribution to approximate binomial probabilities 2 4 6 0 2 n = 10 0 5 10 n = 100 4 6 8 10 n = 30 15 20 10 20 30 40 50 n = 300 ADM2303 - Davood Astaraky Telfer School of Management Shapes of binomial distributions • For this activity you will use a web applet. Example 5 Suppose 35% of all households in Carville have three cars, what is the probabil- We must use a continuity correction (rounding in reverse). SummaryandAp cpati ol i n The previous example suggests the following approximation procedure. The binomial distribution is discrete, and the normal distribution is continuous. Approximating the Binomial distribution Now we are ready to approximate the binomial distribution using the normal curve and using the continuity correction. Under the latter two, this is achieved by showing the convergence, as , of the Laplace or Fourier transform of the Binomial distribution b n p( , ) to a Laplace or Fourier transform, from which then the standard normal distribution is identified as the limiting distribution. Poisson Approximation for the Binomial Distribution • For Binomial Distribution with large n, calculating the mass function is pretty nasty • So for those nasty “large” Binomials (n ≥100) and for small π (usually ≤0.01), we can use a Poisson with λ = nπ (≤20) to approximate it! PROBLEM! Normal approximation to Binomial Review of Normal Distribution Normal approximation 23.3 Normal Distribution Characteristics of the Normal random variable: Let X be a Normal r.v. standard normal probability density function (pdf). Using a normal approximation, the probability that the dice lands on 6 more than 65 times is 0.0438 to 4 decimal places. Find the value of n. (Total for question 2 is 7 marks) 2 A company produces light bulbs. The support for X: (1 ;1) Its parameter(s) and deﬁnition(s): : mean and ˙2: variance The probability density function (pdf): p1 2ˇ˙ e (x )2 2˙2 for 1

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