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central limit theorem, the sample mean is approximately normally distributed. Thus, by the empirical rule, there is roughly a 2.5% chance of being above 54 (2 standard deviations above the mean). (c) Do you need any additional assumptions for part (c) to be true? Solution: No. Since the sample size is large (n 30), the central limit theorem The Central Limit Theorem August 18, 2009 The Normal Distribution If Xis normally distributed with mean and variance ˙2 (we will write this as X˘N( ;˙2)), then its probability density function (pdf) is given by: f(x) = 1 ˙ p 2ˇ e (x )2=2˙2: The graph of this function is bell-shaped, with a maximum at x= , and an approximate width" of 4˙. Central Limit Theorem Formula The central limit theorem is applicable for a sufficiently large sample size (n≥30). The formula for central limit theorem can be stated as follows: Where, μ = Population mean σ = Population standard deviation μ x = Sample mean σ x = Sample standard deviation n = Sample size Applications of Central Limit Theorem Central Limit Theorem General Idea: Regardless of the population distribution model, as the sample size increases, the sample mean tends to be normally distributed around the population mean, and its standard deviation shrinks as n increases. Certain conditions must be met to use the CLT. The samples must be independent and the Central Limit Theorem 6.1 Characteristic Functions 6.1.1 Transforms and Characteristic Functions. There are several transforms or generating functions used in mathematics, prob-abilityand statistics. In general, theyareall integralsof anexponential function, which has the advantage that it converts sums to products. They are all func- THE CENTRAL LIMIT THEOREM normalcdf 85,92,90,p15 25 = 0.6997 Problem 2 Find the average value that is 2 standard deviations above the the mean of the averages. Solution To find the average value that is 2 standard deviations above the mean of the averages, use the formula value =mX+(#ofSTDEVs) psX n value = 90 +2p15 25 •Thep= 1/2 quantile is called the median. •Thep= 1/4 quantile is the first quartile, and thep= 3/4 is called the third quartile. •The interval between the first and third quartiles is called theinterquartile range. •There are also quintiles, and deciles, and, I'm sure, others as well. 12.1.2 Proposition LetFbe the cdf of the random variableX. Central Limit Theorem For real numbers a and b with a b: P a (Xn ) p n ˙ b!! 1 p 2ˇ Z b a e x2=2 dx as n !1. For further info, see the discussion of the Central Limit Theorem in the 10A_Prob_Stat notes on bCourses. Math 10A Law of Large Numbers, Central Limit Theorem be the Brownian local time. Clark-Ocone formula has been used to give a simple proof of the following central limit theorem (Hu-N. '09): Theorem h 3 2 Z R (Lx+h t L x)2dx 4th )8 r t 3 ; as h !0, where t = R R (L x t) 2dx and is a N(0;1) random variable independent of B. David Nualart (Kansas University) Malliavin calculus and CLTs SSP 2017 11 this more general theorem uses the characteristic function (which is deflned for any distribution) `(t) = Z 1 ¡1 eitxf(x)dx = M(it) instead of the moment generating function M(t), where i = p ¡1. Thus the CLT holds for distributions such as the log normal, even though it doesn't have a MGF. Central Limit Theorem 13 The purpose of this problem set is to walk through the proof of the central limit theorem" of probability theory. Roughly speaking, this theorem asserts that if the random variable S nis the sum of many independent random variables S n= X 1+ + X n all with mean zero and nite variance then under appropriate additional hy- potheses 1 s n S n The practical application of this theorem is that, for large n, if Y 1;:::;Y n are indepen-dent with mean y and variance ˙2 y, then Xn i=1 Y i y ˙ y p n! ˘: N(0;1); or Y ˘: N( y;˙2 y=n): How large is large" depends on the distribution of the Y i's. If Normal, then n= 1 is large enough. As the distribution becomes less Normal, larger The practical application of this theorem is that, for large n, if Y 1;:::;Y n are indepen-dent with mean y and variance ˙2 y, then Xn i=1 Y i y ˙ y p n! ˘: N(0;1); or Y ˘: N( y;˙2 y=n): How large is large" depends on the distribution of the Y i's. If Normal, then n= 1 is large enough. As the distribution becomes less Normal, larger

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