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TRANSFORMATION OF FUNCTION OF TWO OR MORE RANDOM VARIABLES

TRANSFORMATION OF FUNCTION OF TWO OR MORE RANDOM VARIABLES. BIVARIATE TRANSFORMATIONS. DISCRETE CASE.

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TRANSFORMATION OF FUNCTION OF TWO OR MORE RANDOM VARIABLES

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  1. TRANSFORMATION OF FUNCTION OF TWO OR MORE RANDOM VARIABLES BIVARIATE TRANSFORMATIONS

  2. DISCRETE CASE • Let X1 and X2 be a bivariate random vector with a known probability distribution function. Consider a new bivariate random vector (U, V) defined by U=g1(X1, X2) and V=g2(X1, X2) where g1(X1, X2) and g2(X1, X2) are some functions of X1 and X2 .

  3. DISCRETE CASE • If B is any subset of 2, then (U,V)B iff (X1,X2)A where • Then, Pr(U,V)B=Pr(X1,X2)A and probability distribution of (U,V) is completely determined by the probability distribution of (X1,X2). Then, the joint pmf of (U,V) is

  4. EXAMPLE • Let X1 and X2 be independent Poisson distribution random variables with parameters 1 and 2. Find the distribution of U=X1+X2.

  5. CONTINUOUS CASE • Let X=(X1, X2, …, Xn) have a continuous joint distribution for which its joint pdf is f, and consider the joint pdf of new random variables Y1, Y2,…, Ykdefined as

  6. CONTINUOUS CASE • If the transformation T is one-to-one and onto, than there is no problem of determining the inverse transformation. An and Bk=n, then T:AB. T-1(B)=A. It follows that there is a one-to-one correspondence between the points (y1, y2,…,yk) in B and the points (x1,x2,…,xn) in A. Therefore, for (y1, y2,…,yk)B we can invert the equation in (*) and obtain new equation as follows:

  7. CONTINUOUS CASE • Assuming that the partial derivatives exist at every point (y1, y2,…,yk=n)B. Under these assumptions, we have the following determinant J

  8. CONTINUOUS CASE called as the Jacobian of the transformation specified by (**). Then, the joint pdf of Y1, Y2,…,Ykcan be obtained by using the change of variable technique of multiple variables.

  9. CONTINUOUS CASE • As a result, the function g is defined as follows:

  10. Example • Recall that I claimed: Let X1,X2,…,Xn be independent rvs with Xi~Gamma(i, ). Then, • Prove this for n=2 (for simplicity).

  11. M.G.F. Method • If X1,X2,…,Xn are independent random variables with MGFs Mxi (t), then the MGF of is

  12. Example • Recall that I claimed: • Let’s prove this.

  13. Example • Recall that I claimed: Let X1,X2,…,Xn be independent rvs with Xi~Gamma(i, ). Then, • We proved this with transformation technique for n=2. • Now, prove this for general n.

  14. More Examples on Transformations • Example 1: • Recall that I claimed: If X~N( , 2), then • Let’s prove this.

  15. Example 2 • Recall that I claimed: Let Xbe an rv with X~N(0, 1). Then, Let’s prove this.

  16. Example 3 Recall that I claimed: • If X and Yhave independentN(0,1) distribution, then Z=X/Yhas a Cauchy distribution with =0 and σ=1. Recall the p.d.f. of Cauchy distribution: Let’s prove this claim.

  17. Example 4 • See Examples 6.3.12 and 6.3.13 in Bain and Engelhardt (pages 207 & 208 in 2nd edition). This is an example of two different transformations: • In Example 6.3.12: In Example 6.3.13: X1 & X2 ~ Exp(1) Y1=X1-X2 Y2=X1+X2 X1 & X2 ~ Exp(1) Y1=X1 Y2=X1+X2

  18. Example 5 • Let X1 and X2 are independent with N(μ1,σ²1) and N(μ2,σ²2), respectively. Find the p.d.f. of Y=X1-X2.

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