Question

prove following

P(x < X ≤ y) = F(y) − F(x)

P(X = x) = F(x) − F(x−) where F(x−) = limy↑x F(y);

P(X = x) = F(x+) − F(x−).

Answer 1

If X be a random variable with probability function P(x) and its distribution function is defined as

F(x)= P(X x)

now 1 P( x<X y) =P(Yy) - P(Xx) = F(y)- F( x) for every x<y

2. F(x) - F( x - ) = P(x-y < X x) since P( Xx) - P(X x- ) = F(x) -F( x- )

3. F(x+ ) - F( x-) = p(x-y < X x+y ) since P( X x+) - P( X x- ) = F(x+) - F(x-)

and P (x- < X < x+ ) = F(x+ ) - F(x-) -- P(X=x) since P(x<X <y) =F(y)- F(x) -- P(X=x)

Hence F( x + ) - F( x -) = P( X=x)

Hence the proof