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SECTION 9 - FUNCTIONS AND TRANSFORMATIONS OF RANDOM VARIABLES

민서 김 English 21 cards

g(u,v) =

Cards in this deck (21)

  1. 1Suppose that X is a continuous random variable with pdf fX(x) and cdf FX(x), and suppose that u(x) is a one-to-one function (usually u is either strictly increasing; e^x, sqrt(x), or lnx, or strictly decreasing; e^-x or 1/x). As a one-to-one function, u has an inverse function v, so that v(u(x)) = x. The random variable Y = u(X) is referred to as atransformation of X.
  2. 2Y = u(X) is a transformation of X. The pdf of Y can be found in one of two ways (they are actually equivalent)i) fY(y) = fX(v(y))*|v'(y)|
    ii) if u is a strictly increasing function, then
    FY(y) = P[Y<=y] = P[u(X)<=y] = P[X<=v(y)] = FX(v(y)), and fY(y) = F'Y(y)
  3. 3Suppose that X is a discrete random variable with probability function f(x). If u(x) is a function of x, and Y is a random variable defined by the equation Y=u(X), then Y is a discrete random variable with probability function g(y) =Given a value of y, find all values of x for which y=u(x) (u(x1)=u(x2)=...=u(xt)=y), and then g(y) is the sum of those f(xi) probabilites.
  4. 4If X and Y are independent random variables, and u and v are functions, then the random variable u(X) and v(Y) areindependent
  5. 5Suppose that the random variables X and Y are jointly distributed with joint density function f(x,y). Suppose also that u and v are functions of the variables x and y. Then U = u(X,Y) and V = v(X,Y) are also random variables with a joint distribution. We wish to find the joint density function of U and V, say g(u,v). We must be able to find inverse functions,h(u,v) and k(u,v) such that x = h(u(x,y),v(x,y)), and y = k(u(x,y),v(x,y))
  6. 6g(u,v) =g(u,v) =
  7. 7Jacobian of the transformationJacobian of the transformation
  8. 8If X1 and X2 are random variables, and Y = X1+X2, then E[Y] =E[X1]+E[X2]
  9. 9If X1 and X2 are random variables, and Y = X1+X2, then Var[Y] =Var[X1]+Var[X2]+2Cov[X1,X2]
  10. 10If X1 and X2 are discrete non-negative integer-valued random variables with joint probability function f(x1,x2), then for an integer k>=0 P[X1+X2=K]If X1 and X2 are discrete non-negative integer-valued random variables with joint probability function f(x1,x2), then for an integer k>=0...
  11. 11If X1 and X2 are discrete non-negative integer-valued independent random variables with f1(x1) and f2(x2), respectively, then P[X1+X2=k] =If X1 and X2 are discrete non-negative integer-valued independent random variables with f1(x1) and f2(x2), respectively, then P[X1+X2=k] =
  12. 12If X1 and X2 are continuous random variables with joint density function f(x1,x2) then the density function of Y = X1+X2 is fY(y) =If X1 and X2 are continuous random variables with joint density function f(x1,x2) then the density function of Y = X1+X2 is fY(y) =
  13. 13If X1 and X2 are independent continuous random variables with density functions f1(x1) and f2(x2), then the density function of Y=X1+X2 is fY(y) =If X1 and X2 are independent continuous random variables with density functions f1(x1) and f2(x2), then the density function of Y=X1+X2 i...
  14. 14If X1 and X2 are independent continuous random variables with density functions f1(x1) and f2(x2), X1>=0 and X2>=0, then the density function of Y=X1+X2 is fY(y) =If X1 and X2 are independent continuous random variables with density functions f1(x1) and f2(x2), X1>=0 and X2>=0, then the density func...
  15. 15If X1,X2,...,Xn are random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then E[Y] =If X1,X2,...,Xn are random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then E[Y] =
  16. 16If X1,X2,...,Xn are random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then Var[Y] =If X1,X2,...,Xn are random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then Var[Y] =
  17. 17If X1,X2,...,Xn are mutually independent random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then Var[Y] =If X1,X2,...,Xn are mutually independent random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then Var[...
  18. 18If X1,X2,...,Xn are mutually independent random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then MY(t) =If X1,X2,...,Xn are mutually independent random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then MY(t) =
  19. 19If X1,X2,...,Xn are mutually independent random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then PY(t) =If X1,X2,...,Xn are mutually independent random variables, and the random variable Y is defined to be Y = (sum from i=1 to n)Xi then PY(t) =
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Flashcards on SECTION 9 - FUNCTIONS AND TRANSFORMATIONS OF RANDOM VARIABLES, created by 민서 김 on 08/02/2022.
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