Finding the Probability and cumulative distribution of a Random Variable.

Question

Find the distribution (for real a find (P({ω:X(ω)=a})) and the cumulative distribution function of X defined on Ω_{coin toss} by X(Heads)=1, X(Tails)=-1 (for each real a find the value of F_x(a)=P({ω:X(ω)≤a}))

 

I am having trouble with being able to understand this question let alone answer it if there is anyone that can help.

Answer 1

This problem is presented as one would expect from a University course on probability.

 

Remember that F(a) = P( X<=a)  is monotonically non decreasing. there are steps in the function at a = -1 and a = 1.

 

for a < -1, F(a) = P(X<=a) = 0

 

At a = -1, F(a) = P(X<= -1) = p (provided p= prob of tails)

 

for -1< x <= a, for a < 1, F(a) = p

 

for 1<=a, F(a) = 1 provided prob of head or tails is 1