Math Is Fun Forum

abc4616

Member

Registered: 2006-10-01

Posts: 9

Cumulative Distribution Question

The C.D.F. (cumulative density function) of the random variable X is given by,

F(x)= [0,  x<0                               
          x/2,  0<=x<1
          2/3, 1<=x<2
          11/12, 2<=x<3
          1,  3<=x]

a) Find P(X>1/2)

b) Find P(2<X<=4)

c) Find P(X=1)

Does anyone know how to approach this?

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: Cumulative Distribution Question

Recall the definition (note that X is a random variable and x is the argument to the function):

the probability P[X <= x] = F(x)

We know F(x).

a) P[X > 1/2] = 1 - P[X <= 1/2]

now use the definition of F(x):

P[X <= 1/2] = F(1/2) = (1/2)/2 = 1/4

since this is F(x) for 0 <= x < 1

So now,

P[X > 1/2] = 1 - P[X <= 1/2] = 1 - 1/4 = 3/4

I don't quite remember how to do the others at this moment, hopefully other posters can help!

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: Cumulative Distribution Question

(b)

Here I think you need the formula:

P[a < X <= b] = F(b) - F(a)

so

P[2 < X <= 4] = F(4) - F(2) = 1 - 11/12 = 1/12