18.440: Lecture 19 Normal random variables
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1 Lecture : Lecture 19 Normal random variables Scott Sheffield MIT
2 Outline Tossing coins Normal random variables Special case of central limit theorem
3 Outline Tossing coins Normal random variables Special case of central limit theorem
4 Lecture 19 Tossing coins Suppose we toss a million fair coins. How many heads will we get?
5 Lecture 19 Tossing coins Suppose we toss a million fair coins. How many heads will we get? About half a million, yes, but how close to that? Will we be off by 10 or 1000 or 100,000?
6 Tossing coins Suppose we toss a million fair coins. How many heads will we get? About half a million, yes, but how close to that? Will we be off by 10 or 1000 or 100,000? How can we describe the error?
7 Tossing coins Suppose we toss a million fair coins. How many heads will we get? About half a million, yes, but how close to that? Will we be off by 10 or 1000 or 100,000? How can we describe the error? Let s try this out.
8 Tossing coins Toss n coins. What is probability to see k heads?
9 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k).
10 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k). Let s plot this for a few values of n.
11 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k). Let s plot this for a few values of n. Seems to look like it s converging to a curve.
12 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k). Let s plot this for a few values of n. Seems to look like it s converging to a curve. If we replace fair coin with p coin, what s probability to see k heads.
13 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k). Let s plot this for a few values of n. Seems to look like it s converging to a curve. If we replace fair coin with p coin, what s probability to see k heads. Answer: p k (1 p) n k( n k).
14 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k). Let s plot this for a few values of n. Seems to look like it s converging to a curve. If we replace fair coin with p coin, what s probability to see k heads. Answer: p k (1 p) n k( n k). Let s plot this for p = 2/3 and some values of n.
15 Tossing coins Toss n coins. What is probability to see k heads? Answer: 2 k( n k). Let s plot this for a few values of n. Seems to look like it s converging to a curve. If we replace fair coin with p coin, what s probability to see k heads. Answer: p k (1 p) n k( n k). Let s plot this for p = 2/3 and some values of n. What does limit shape seem to be?
16 Outline Tossing coins Normal random variables Special case of central limit theorem
17 Outline Tossing coins Normal random variables Special case of central limit theorem
18 Standard normal random variable Say X is a (standard) normal random variable if f X (x) = f (x) = 1 e x2 /2.
19 Standard normal random variable Say X is a (standard) normal random variable if f X (x) = f (x) = 1 e x2 /2. Clearly f is always non-negative for real values of x, but how do we show that f (x)dx = 1?
20 Standard normal random variable Say X is a (standard) normal random variable if f X (x) = f (x) = 1 e x2 /2. Clearly f is always non-negative for real values of x, but how do we show that f (x)dx = 1? Looks kind of tricky.
21 Standard normal random variable Say X is a (standard) normal random variable if f X (x) = f (x) = 1 e x2 /2. Clearly f is always non-negative for real values of x, but how do we show that f (x)dx = 1? Looks kind of tricky. Happens to be a nice trick. Write I = e x2 /2 dx. Then try to compute I 2 as a two dimensional integral.
22 Standard normal random variable Say X is a (standard) normal random variable if f X (x) = f (x) = 1 e x2 /2. Clearly f is always non-negative for real values of x, but how do we show that f (x)dx = 1? Looks kind of tricky. Happens to be a nice trick. Write I = e x2 /2 dx. Then try to compute I 2 as a two dimensional integral. That is, write I 2 = e x2 /2 dx e y 2 /2 dy = e x2 /2 dxe y 2 /2 dy.
23 Standard normal random variable Say X is a (standard) normal random variable if f X (x) = f (x) = 1 e x2 /2. Clearly f is always non-negative for real values of x, but how do we show that f (x)dx = 1? Looks kind of tricky. Happens to be a nice trick. Write I = e x2 /2 dx. Then try to compute I 2 as a two dimensional integral. That is, write I 2 = e x2 /2 dx e y 2 /2 dy = Then switch to polar coordinates. I 2 = 0 0 so I =. e r 2 /2 rdθdr = 0 e x2 /2 dxe y 2 /2 dy. re r 2 /2 dr = e r 2 /2, 0
24 Standard normal random variable mean and variance Say X is a (standard) normal random variable if f (x) = 1 e x2 /2.
25 Standard normal random variable mean and variance Say X is a (standard) normal random variable if f (x) = 1 e x2 /2. Question: what are mean and variance of X?
26 Standard normal random variable mean and variance Say X is a (standard) normal random variable if f (x) = 1 e x2 /2. Question: what are mean and variance of X? E[X ] = xf (x)dx. Can see by symmetry that this zero.
27 Standard normal random variable mean and variance Say X is a (standard) normal random variable if f (x) = 1 e x2 /2. Question: what are mean and variance of X? E[X ] = xf (x)dx. Can see by symmetry that this zero. Or can compute directly: E[X ] = 1 e x2 /2 xdx = 1 e x2 /2 = 0.
28 Standard normal random variable mean and variance Say X is a (standard) normal random variable if f (x) = 1 e x2 /2. Question: what are mean and variance of X? E[X ] = xf (x)dx. Can see by symmetry that this zero. Or can compute directly: E[X ] = 1 e x2 /2 xdx = 1 e x2 /2 = 0. How would we compute Var[X ] = f (x)x 2 dx = 1 e x2 /2 x 2 dx?
29 Lecture 19 Standard normal random variable mean and variance Say X is a (standard) normal random variable if f (x) = 1 e x2 /2. Question: what are mean and variance of X? E[X ] = xf (x)dx. Can see by symmetry that this zero. Or can compute directly: E[X ] = 1 e x2 /2 xdx = 1 e x2 /2 = 0. How would we compute Var[X ] = f (x)x 2 dx = 1 e x2 /2 x 2 dx? Try integration by parts with u = x and dv = xe x2 /2 dx. Find that Var[X ] = 1 ( xe x2 /2 + e x2 /2 dx) = 1.
30 Lecture 19 General normal random variables Again, X is a (standard) normal random variable if f (x) = 1 e x2 /2.
31 Lecture 19 General normal random variables Again, X is a (standard) normal random variable if f (x) = 1 e x2 /2. What about Y = σx + µ? Can we stretch out and translate the normal distribution (as we did last lecture for the uniform distribution)?
32 Lecture 19 General normal random variables Again, X is a (standard) normal random variable if f (x) = 1 e x2 /2. What about Y = σx + µ? Can we stretch out and translate the normal distribution (as we did last lecture for the uniform distribution)? Say Y is normal with parameters µ and σ 2 if f (x) = 1 σ e (x µ)2 /2σ 2.
33 Lecture 19 General normal random variables Again, X is a (standard) normal random variable if f (x) = 1 e x2 /2. What about Y = σx + µ? Can we stretch out and translate the normal distribution (as we did last lecture for the uniform distribution)? Say Y is normal with parameters µ and σ 2 if f (x) = 1 σ e (x µ)2 /2σ 2. What are the mean and variance of Y?
34 Lecture 19 General normal random variables Again, X is a (standard) normal random variable if f (x) = 1 e x2 /2. What about Y = σx + µ? Can we stretch out and translate the normal distribution (as we did last lecture for the uniform distribution)? Say Y is normal with parameters µ and σ 2 if f (x) = 1 σ e (x µ)2 /2σ 2. What are the mean and variance of Y? E[Y ] = E[X ] + µ = µ and Var[Y ] = σ 2 Var[X ] = σ 2.
35 Lecture 19 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2.
36 Lecture 19 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2. What is the cumulative distribution function?
37 Lecture 19 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2. What is the cumulative distribution function? Write this as F X (a) = P{X a} = 1 a e x2 /2 dx.
38 Lecture 19 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2. What is the cumulative distribution function? Write this as F X (a) = P{X a} = 1 a e x2 /2 dx. How can we compute this integral explicitly?
39 Lecture 19 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2. What is the cumulative distribution function? Write this as F X (a) = P{X a} = 1 a e x2 /2 dx. How can we compute this integral explicitly? Can t. Let s just give it a name. Write Φ(a) = 1 a e x2 /2 dx.
40 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2. What is the cumulative distribution function? Write this as F X (a) = P{X a} = 1 a e x2 /2 dx. How can we compute this integral explicitly? Can t. Let s just give it a name. Write Φ(a) = 1 a e x2 /2 dx. Values: Φ( 3).0013, Φ( 2).023 and Φ( 1).159.
41 Cumulative distribution function Again, X is a standard normal random variable if f (x) = 1 e x2 /2. What is the cumulative distribution function? Write this as F X (a) = P{X a} = 1 a e x2 /2 dx. How can we compute this integral explicitly? Can t. Let s just give it a name. Write Φ(a) = 1 a e x2 /2 dx. Values: Φ( 3).0013, Φ( 2).023 and Φ( 1).159. Rough rule of thumb: two thirds of time within one SD of mean, 95 percent of time within 2 SDs of mean.
42 Outline Tossing coins Normal random variables Special case of central limit theorem
43 Outline Tossing coins Normal random variables Special case of central limit theorem
44 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin.
45 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin. What s the standard deviation of S n?
46 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin. What s the standard deviation of S n? Answer: npq (where q = 1 p).
47 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin. What s the standard deviation of S n? Answer: npq (where q = 1 p). The special quantity Sn np npq describes the number of standard deviations that S n is above or below its mean.
48 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin. What s the standard deviation of S n? Answer: npq (where q = 1 p). The special quantity Sn np npq describes the number of standard deviations that S n is above or below its mean. What s the mean and variance of this special quantity? Is it roughly normal?
49 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin. What s the standard deviation of S n? Answer: npq (where q = 1 p). The special quantity Sn np npq describes the number of standard deviations that S n is above or below its mean. What s the mean and variance of this special quantity? Is it roughly normal? DeMoivre-Laplace limit theorem (special case of central limit theorem): lim P{a S n np n npq b} Φ(b) Φ(a).
50 Lecture 19 DeMoivre-Laplace Limit Theorem Let S n be number of heads in n tosses of a p coin. What s the standard deviation of S n? Answer: npq (where q = 1 p). The special quantity Sn np npq describes the number of standard deviations that S n is above or below its mean. What s the mean and variance of this special quantity? Is it roughly normal? DeMoivre-Laplace limit theorem (special case of central limit theorem): lim P{a S n np n npq b} Φ(b) Φ(a). This is Φ(b) Φ(a) = P{a X b} when X is a standard normal random variable.
51 Lecture 19 Problems Toss a million fair coins. Approximate the probability that I get more than 501, 000 heads.
52 Lecture 19 Problems Toss a million fair coins. Approximate the probability that I get more than 501, 000 heads. Answer: well, npq = = 500. So we re asking for probability to be over two SDs above mean. This is approximately 1 Φ(2) = Φ( 2).159.
53 Problems Toss a million fair coins. Approximate the probability that I get more than 501, 000 heads. Answer: well, npq = = 500. So we re asking for probability to be over two SDs above mean. This is approximately 1 Φ(2) = Φ( 2).159. Roll dice. Expect to see sixes. What s the probability to see more than 9800?
54 Problems Toss a million fair coins. Approximate the probability that I get more than 501, 000 heads. Answer: well, npq = = 500. So we re asking for probability to be over two SDs above mean. This is approximately 1 Φ(2) = Φ( 2).159. Roll dice. Expect to see sixes. What s the probability to see more than 9800? Here npq =
55 Problems Toss a million fair coins. Approximate the probability that I get more than 501, 000 heads. Answer: well, npq = = 500. So we re asking for probability to be over two SDs above mean. This is approximately 1 Φ(2) = Φ( 2).159. Roll dice. Expect to see sixes. What s the probability to see more than 9800? Here npq = And 200/ Answer is about 1 Φ( 2.19).
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