The story of the film so far... Mathematics for Informatics 4a. Jointly distributed continuous random variables. José Figueroa-O Farrill

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1 The story of the film so far... Mathematics for Informatics 4a José Figueroa-O Farrill Lecture 2 2 March 22 X a c.r.v. with p.d.f. f and g : R R: then Y g(x is a random variable and E(Y g(f(d variance: Var(X E(X 2 E(X 2 moment generating function: M X (t E(e tx have met uniform, eponential and normal distributions and have computed their mean, variance and m.g.f. if X normally distributed with mean µ and variance σ 2, Y σ (X µ has standard normal distribution The c.d.f. Φ of the standard normal distribution is not an elementary function, but there are tables maimum entropy: normal distribution is the least biased among all p.d.f.s with the same mean and variance José Figueroa-O Farrill mi4a (Probability Lecture 2 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 2 / 2 Jointly distributed continuous random variables Two continuous random variables X and Y are said to be jointly distributed with joint density f(, y if for all a < b and c < d, P(a < X < b, c < Y < d d b c a f(, yddy It follows that the joint density obeys f(, y and Eample Let X and Y have joint density f(, y cy, y. What is c? From the normalisation condition, ( ( cy d dy c y2 c 4 c 4 and (provided C is nice enough f(, yddy P((X, Y C f(, yddy C What if < y? Since the density is symmetric in y, the integral over half the square is half of the previous result, hence c is twice the previous value: c 8. José Figueroa-O Farrill mi4a (Probability Lecture 2 3 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 4 / 2

2 Uniform joint densities Let A R 2 be a region with area A. X and Y are (jointly uniform in A if f(, y { A, (, y A, elsewhere Marginals Let X, Y be continuous random variables with joint density f(, y. Then the marginal p.d.f.s f X ( and f Y (y are given by f X ( f(, ydy and f Y (y f(, yd Eample Let X, Y be jointly uniform in the unit disk D {(, y 2 + y 2 }. Then D π, whence f(, y π 2 + y 2 Remark As in the discrete case there is no need to stop at two random variables, and we can have joint densities f(,..., n for n jointly distributed random variables, with many different marginals. José Figueroa-O Farrill mi4a (Probability Lecture 2 5 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 6 / 2 Eample Let X, Y be jointly uniform on the unit disk D: f(, y 2 + y 2 π Joint distributions Let X and Y be continuous random variables with joint density f(, y. Their joint distribution is defined as The marginals are given by F(, y P(X, Y y y f(u, vdu dv y 2 2 f X ( 2 π dy 2 2 π It follows from the fundamental theorem of calculus that 2 for and, by symmetry, f Y (y 2 π y 2 f(, y 2 F(, y y and the marginal distributions are obtained by for y F X ( F(, and F Y (y F(, y José Figueroa-O Farrill mi4a (Probability Lecture 2 7 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 8 / 2

3 Eample Let X, Y be jointly distributed with f(, y + y on, y. One checks that indeed ( + yddy. The joint distribution is F(, y y (u + vdu dv ( y (u + vdv du ( uy + 2 y2 du 2 2 y + 2 y2 for, y For y >, F(, y 2( + and similarly, for >, F(, y 2 y(y +. Independence Two continuous random variables X and Y are independent if or, equivalently, F(, y F X (F Y (y f(, y f X (f Y (y. It follows that for X, Y independent P(X A, Y B P(X AP(Y B Useful criterion: X and Y are independent iff f(, y g(h(y. Then f X ( cg( and f Y (y c h(y, where c R h(ydy. José Figueroa-O Farrill mi4a (Probability Lecture 2 9 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 / 2 Eamples X and Y are jointly uniform on a and y b: f(, y ab for (, y [, a] [, b] with marginals f X ( a and f Y(y b. Since f(, y f X (f Y (y, X and Y are independent. 2 X and Y are jointly uniform on the disk 2 + y 2 a 2 : f(, y πa 2 for 2 + y 2 a 2 with marginals f X ( πa a 2 2 and 2 f Y (y πa a 2 y 2. Since f(, y f 2 X (f Y (y, X and Y are not independent. Geometric probability Geometric probability or continuous combinatorics studies geometric objects sharing a common probability space. We have already seen some geometric probability problems in the tutorial sheets. For eample, in Tutorial Sheet 4 you considered the problem of tossing a coin on a square grid and computing the probability that the coin is fully contained inside one of the squares. This game was called franc-carreau ( free tile in France and was studied by Buffon in his treatise Sur le jeu de franc-carreau (733. In probability, Buffon is perhaps better known for Buffon s needle, which is a paradigmatic geometric probability problem. José Figueroa-O Farrill mi4a (Probability Lecture 2 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 2 / 2

4 Buffon s needle I Drop a needle of length l at random on a striped floor, with stripes a distance L apart. Let l < L: short needles. L l What is the probability that the needle does not touch any line? Buffon s needle II The needle is described by the midpoint and the angle with the horizontal. Symmetry allows us to ignore the vertical component of the midpoint and to assume the horizontal component lies in one of the strips. Let X denote the horizontal component of the midpoint. It is uniformly distributed in [ L 2, L 2 ]. Let Θ denote the angle with the horizontal, which is uniformly distributed in [ π 2, π 2 ]. Since X and Θ are independent, the joint probability density function is the product of the two probability density functions and hence is also uniformly distributed. θ José Figueroa-O Farrill mi4a (Probability Lecture 2 3 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 4 / 2 Buffon s needle III The needle will touch one of the parallel lines if and only if + l 2 cos θ > L 2 for [ L 2, L 2 ] and θ [ π 2, π 2 ]. The complementary probability is P ( X 2 (L l cos Θ [ π 2 2 (L l cos θ π 2 π 2 π 2 π 2 l 2 (L l cos θ d (L l cos θdθ π 2 ] θ dθ cos θdθ 2l Functions of several random variables X and Y are continuous random variables with joint density f(, y Z g(x, Y, for some function g : R 2 R How is Z distributed? (assuming it is a c.r.v. Its c.d.f. F Z (z P(Z z is given by F Z (z f(, yd dy Its p.d.f. F Z (z g(,yz José Figueroa-O Farrill mi4a (Probability Lecture 2 5 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 6 / 2

5 Eample (The sum of two jointly uniform variables Let X and Y be jointly uniform on [, ] f(, y f X ( f Y (y for, y, so X and Y are independent. Let Z X + Y. F Z (z +yz d dy { 2 z 2, z [, ] 2 (2 z2, z [, 2] z y 2 z The sum of two independent variables: convolution Let X and Y be continuous random variables with joint density f(, y and let Z X + Y. Then F Z (z +yz f(, yd dy is given by ( z F Z (z f(, ydy d Hence F Z (z is given by f(, z d { z, z [, ] 2 z, z [, 2] 2 z If X and Y be independent, f(, y f X (f Y (y, whence f X (f Y (z d (f X f Y (z which defines the convolution product José Figueroa-O Farrill mi4a (Probability Lecture 2 7 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 8 / 2 Convolution The convolution product satisfies a number of interesting properties: commutativity: f g g f associativity: (f g h f (g h smoothing: f g is a smoother function than f or g, e.g., Summary C.r.v.s X and Y have a joint density f(, y with P((X, Y C f(, yd dy and a joint distribution C F(, y P(X, Y y y with f(, y 2 yf(, y X and Y independent iff f(, y f X (f Y (y Geometric probability is fun! (Buffon s needle f(u, vdu dv We can calculate the c.d.f. and p.d.f. of Z g(x, Y X, Y independent: f X+Y f X f Y (convolution José Figueroa-O Farrill mi4a (Probability Lecture 2 9 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 2 / 2

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