5.6 The Normal Distributions

Transcription

1 STAT 41 Lecture Notes The Normal Distributions Definition A (continuous) random variable X has a normal distribution with mean µ R and variance < R if the p.d.f. of X is f(x µ, ) ( π ) 1/ 1 ( ) ] x µ, (1) for x R. Theorem The function given in formula (1) is a p.d.f. To prove the claim, first note that < f(x µ, ) for all x R, < and µ R. To establish 1 f(x µ, ), set so that dy and y x µ I 1 1 ( ) ] x µ y dy. Theorem is true if I π. To prove I π, note that I I I z A change of variables to polar coordinates is helpful: y dydz 1 ( y + z )] dydz. y r cos(θ) z r sin(θ) r y + z. The Jacobian of the transformation is ( J det cos θ r sin θ sin θ r cos θ ) r.

2 STAT 41 Lecture Notes 133 Then, I π π π 1 ( y + z )] dydz r r drdθ r dθ dθ π. Hence, f(x µ, ) is a p.d.f. The m.g.f. of X N(µ, ) is ] ψ(t) (π ) 1/ (x µ) tx ] (π ) 1/ tx (x xµ + µ ) ) ] ( µ (π ) 1/ x x(µ + t) ( ) (µ + t) ] µ (π ) 1/ x x(µ + t) + (µ + t) (µt + 1 ) t. Then, E(X) ψ () ψ() ( µ + ) µ. E(X ) d ψ(t) (µ + t)] dt ψ(t) ( µ + t ) + ψ(t) µ + Var(X). Since f(x µ, ) is symmetric, all odd moments are. Consider logf(x µ, )] 1 ( ) ] x µ log(π ) + ; the derivative of logf(x µ, )] with respect to x is d logf(x µ, )] 1 ( ) x µ.

3 STAT 41 Lecture Notes 134 Since log is monotonically increasing (and hence, order-preserving), arg max x f(x µ, ) ] { arg max logf(x µ, )] } µ. x The p.d.f. has two inflection points that can be determined by finding x such that d f(x µ, ). After some algebra, d f(x µ, ) f(x µ, ) + 4 (x µ) ]. Setting d f(x µ, ) and solving for x shows that there are inflection points at µ and µ +. The figures below show the graph of f(x, 1) and f(x µ 5, 1). Note that the shapes are identical Theorem If X has a normal distribution with mean µ and variance, and Y ax +b, where a, then Y has a normal distribution with mean aµ + b and variance a. The proof is straightforward: ψ Y (t) E e t(ax+b)]] e bt ψ X (at) (bt + aµt + 1 ) a t (b + aµ)t + 1 ] a t,

4 STAT 41 Lecture Notes 135 which is recognizable as the m.g.f. variance a. of a normal random variable with mean aµ + b and The Standard Normal Distribution Suppose that X N(, 1). Then, Z is said to have a standard normal distribution. Commonly, the notation ϕ(x) is used to denote the p.d.f. and Φ(x) x ϕ(t) dt is used to denote the c.d.f. A closed-form ression for Φ(x) is unavailable. Because of symmetry of ϕ(x) about, there are several convenient relationships of note: ϕ(x) 1 ϕ(x) Φ 1 (p) Φ 1 (1 p) Pr( x X x) Φ(x) Φ( x) 1 Φ( x). Traditionally, standard normal tables were used to determine probabilities involving X N(µ, ). 1 Using the result of Theorem , these probabilities are approximated by transforming X to a standard normal random variable so that the standard normal tables could be used. For example, ( X µ Pr(X x) Pr Pr(Z z), where z x µ. Then, Φ(z) is extensively tabled for 3.8 z 3.8. x µ ) ( ) x µ Pr(X x) Φ. Some quantiles of the N(µ, ) distribution are also computable using normal tables. Let x p denote the pth percentile of the N(µ, ) distribution. First note that given p, x p satisfies p Pr(X x p ). Find the pth percentile of the standard normal distribution in the standard normal table, i.e., z p satisfying p. Φ(z p ). Since ( ) xp µ. p Pr(X x p ) Φ Φ(zp ), 1 Use of the standard normal tables are still taught to students in less-developed countries.

5 STAT 41 Lecture Notes 136 set z p x p µ and solve for x p. The pth percentile is (approximately) x p µ + z p. A useful property of the normal distributions is that for all normal random variables X N(µ, ) and k, Pr ( X µ k) Pr ( Z k) ; in other words, the probability that a normal random variable (with mean µ and variance ) will be within k of µ is the same for all permissible values of µ and. It s useful then to find probabilities after partitioning the domain of a standard normal into intervals defined by µ ±, µ ± and µ ± 3. The partitioning of total probability is shown in the figure below % 68% 13.5% 13.5%.35% 4 4 z It s not shown, but Pr(Z 3).15. Linear combinations of normally distributed random variables Theorem Suppose that X i N(µ i, i ), i 1,..., k are independent. Then ( Xi N µi, ) i.

6 STAT 41 Lecture Notes 137 A proof using m.g.f. s is straightforward, so let ψ i (t) denote the m.g.f. of X i. By independence, the m.g.f. of X i is ψ(t) ψ i (t) ( µ i t + 1 t i ( t µ i + t ) i ), which is the m.g.f. of a random variable distributed as N ( µ i, i ). Theorem is often applied when probabilities are wanted involving the difference between two normal random variables. Y N(µ Y, Y ). For example, suppose that X N(µ X, X ) and Then Pr(X < Y ) is easily computed by setting D Y X so that Pr(X < Y ) Pr(D > ). Since D N(µ Y µ X, X + Y ), computing Pr(D > ) is trivial using the R function pn. There are a variety of applets available on-line as well. Corollary Suppose that X i N(µ i, i ), i 1,..., k are independent and a 1,..., a k are constants, of which at least one is non-zero. Then ( ai X i + b N ai µ i + b, ) a i i. Corollary Suppose that X i N(µ, ), i 1,..., k are a random sample from X N(µ, ). Then X n n 1 X i N(µ, /n). Example To compute Pr ( X n µ < k ), for some k >, one may proceed as follows: Pr ( X n µ < k ) ( ) X n µ Pr / nk n < ( ) nk nk Pr < Z < ( ) nk 1 Φ. To find the smallest n such that Pr ( X n µ < k ) p, first find z p such that p 1 Φ ( z p ) (use the R quantile function qn). For example, if p.95, then z p Finally, solve for n using the equation z p nk.

7 STAT 41 Lecture Notes 138 Lognormal Distributions It s sometimes the case that X has a distribution that is right-skewed with support on (, ) or some subset thereof. In these cases, the transformation Y log(x) often has a distribution that appears to be approximately normal, and the normal distribution will may be used to conduct inference on log(x). Since the log function is a monotone transformation, inferences about log(x) carry over, at least in part, to X. For instance, suppose that Y log(x) N(µ, ). as follows: To compute Pr(X x), proceed Pr(X x) Pr(e Y x) PrY log(x)] Pr Z log(x) µ ]. Similarly, if the pth percentile of the distribution of X is desired (call it x p ), we may proceed as follows: p Pr(X x p ) Pr(e Y x p ) Pr Y log(x p )] ] log(xp ) µ Φ. Find the pth quantile of the standard normal distribution, and call it z p. Solve for x p : z p log(x p) µ x p (µ + z p ). If log(x) N(µ, ), then X is said to have a lognormal distribution with parameters µ and. We my find the p.d.f. of X as follows: F X (x) Pr ( e Y x ) Pr (Y log(x)) F Y log(x)] f X (x) f Y log(x)] d log(x) { (π ) 1/ x 1 1 } log(x) µ] I (, ) (x). Inference is the process of drawing conclusions about X, or an underlying population which generates X via a procedure such as random sampling from a set of realizations of X.