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AMS 311, Lecture 14 March 27, 2001 Two problem quiz next class (March 29): univariate transformation of a random variable and moment of a univariate random variable. You may use one sheet of notes. Homework for Chapter Seven (due April 5): Starting on page 252: 6, 10, 14*; starting on page 266: 8, 16, 24*; starting on page 274: 4, 8; starting on page 285: 2, 8, 12. Theorem 6.1. (Method of Transformations) Let X be a continuous random variable with density function fX and the set of possible values A. For the invertible function h: AR, let Y=h(X) be a random variable with the set of possible values B=h(A)={h(a):aA}. Suppose that the inverse of y=h(x) is the function x=h-1(y), which is differentiable for all value of yB. Then fY, the density function of Y, is given by f Y ( y) f X (h 1 ( y))|(h 1 )'( y)|, y B. In computer simulation, one applied the probability integral transformation to generate values following a specified distribution. Two example density function of a random variable problems: Example 1. Let U be a uniform(0,1) random variable. Find the distribution of Y=-ln(1-U). Always look for two ways. Direct: Find the cdf of Y : FY ( y) P(Y y) P( ln(1 U ) y) P(1 U e y ) P(U 1 e y ). From this one can find the pdf by differentiation. Use of Theorem 6.1. The inverse function is u 1 e y . Hence the differential element du e y dy. Definition of Expected Value If X is a continuous random variable with probability density function f, the expected value of X is defined by E ( X ) xf ( x)dx, provided that the integral converges absolutely. Definition of var (X): The variance of the random variable X is still var( X ) E ( X EX ) 2 . Example c , x , is called a 1 x2 Cauchy random variable. Find c so that the f(x) is a pdf. Show that E(X) does not exist. A random variable X with density function f ( x ) Don’t be bashful about checking your old calculus books and tables of integrals! From there, you will find dx 1 x 2 arctan x. Law of the unconscious statistician (I prefer to call this the law of the choice of probability measures). Theorem 6.3. Let X be a continuous random variable with probability density function f(x); then for any function h: RR, E (h( X )) h( x) f ( x)dx. Uniform distribution: A random variable X is uniformly distributed over the interval (a, b) if its pdf is 1 f ( x) , a x b, and zero otherwise. b a ab (b a ) 2 , and var( X ) Then E ( X ) . 2 12 Example. Let the random variable X be uniform (0,1). Find E (e tX ). The function in the last problem is extremely important in later chapters. It is called the moment generating function. Normal Distribution Statement of pdf. The cdf is tabulated and is a basic reference for working problems. De Moivre’s Theorem: Central limit theorem type result for approximating number of heads in n independent tosses of a fair coin. De Moivre-Laplace Theorem. Generalization to n independent Bernoulli trials with probability of success p. All probabilities are calculated through conversion to a standard normal distribution. Basic principle of p-values in statistical tests.