Lecture 16

1. Lecture 16 Description of random variables: pdf, cdf. Expectation. Variance.

2. Review: Area under a curve Approximate by rectangles. Width  0, areas of rectangles  integral. Riemann integral, by definition, is the limit of such approximations

3. Mass of rod with uneven density • Density (x), is a function of x. • The mass between x1 and x2, is the definite integral of (x) from x1 to x2. • If the difference (x2 - x1) is so small such that the density is roughly constant, the mass between x1 and x2 is approximately (x1)(x2 - x1). x

4. M is the total mass. Center of Gravity string • Center of mass, or center of gravity, is the point where we can hold the rod in equilibrium (an unstable equilibrium).

5. Moment of Inertia • The second moment also have mechanical meaning. M is the total mass.

6. Simulation of continuous by digital computer • is never exact. • provides good approximation if precision is high enough.

7. Mass vs Probability • Probability density function can be viewed as the density of a rod with unit mass. • Mass of a single point with zero length is zero. • The “center of gravity” of a pdf is the expectation of the random variable. • The “moment of inertia” is the second moment.

8. Cumulative density function of discrete random variable • F(x) =def P(Xx). Example: Bernoulli random variable P(X=0) = p, P(X=1)=1-p. F(x) 1 p x 1

9. f(x) = p(x)+(1-p) (x-1) 0 1 Derivative? • Derivative of discontinuous function does not exist. • However, if we allow generalized functions, such as the unit impulse function, also known as Dirac delta function, than we can talk about the pdf of discrete random variables. • The pdf of Bernoulli random variable is the sum of two delta functions.

10. Cumulative density function of continuous random variable • Cdf F(x) of continuous random variable are differentiable (or piecewise differentiable). • The derivative f(x) of cdf is the pdf. • Fundamental theorem of calculus: (It is also equal to P(a<Xb) as P(X=a)=0.)

11. Example: Uniform r.v. F(x) 1 • Cdf and pdf carry the same information. • Usually we use pdf in computation. • Cdf are found in tables of probability distribution. 1 x f(x) 1 x 1

12. How to generate random variable? • If the cdf has a simple form, then we can use the inverse transform method. • To generate a random variable with cdf F(x) • Compute the inverse of F. • Generate a uniform random variable U between 0 and 1. • Return F-1(U).

13. Expectation • Suppose the pdf of a random variable X is fX(x). The expectation of X is defined as The integral reduces to summation if X is a discrete random variable, i.e., when fX(x) is a sum of delta functions.

14. Easy properties • E[X+b]=E[X]+b. • E[aX] = aE[X]. • E[g(X)]g(E[X]) in general. • If g(x)h(x) for all x, than E[g(X)] E[h(X)].

15. Second moment and variance • Second moment of a random variable X is the expectation of X2. • E[X2] • Variance measures the level of variation from the mean. It is defined as the expectation of the square of deviation from the mean, • Var(X)=E[(X-E[X])2].

16. Properties • Var(X) = E[X2]-(E[X])2. • Var(X+b) = Var(X). • Var(aX) = a2 Var(X).

17. What is the pdf of a function of X? • Let fX(x) be the pdf of random variable X. • What is the pdf of • X+b? • aX? • X2? • g(X), where g(x) is a monotonically increasing function?