Let X1 and X2 be independent random variables taking values

Let X1 and X2 be independent random variables taking values in some set with finite cardinality and let Y1 = g1(X1) and Y2 = g2(X). (a) Show that P(Y1 = r, Y2 = s) = sum P(X1 = a, X2 = b) (b) Using (a), show that P(Y1 = r, Y2 = s) = P(Y1 = r)P(Y2 = s) so that Y1 and Y2 are independent.

Solution

Y1 = g1(x1) and Y2 = g2(x2)

a) P(Y1=r, Y2 =s) = P{g1(x1) =r, g2(x2) = s}

= Sum of all probabilities such that g1(a) =r , g2(b) =s where x1 =a and x2 =b

Thus proved

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b) The summation nth term is P(x1=a, x2 =b)

As x1 and x2 are independent, this = P(x1=a)P(x2=b)

Hence P(Y1=r, Y2 =s) = P{g1(x1) =r, g2(x2) = s}

= P(y1=r)(P(y2=s)

Or y1 and y2 are independent.