# Probability of continuous random variables

1. Nov 16, 2007

### Gott_ist_tot

1. The problem statement, all variables and given/known data
A random variable has distribution function F(z) = P(y<= z) given by (this is a piecewise function)

f(z) =
0 if z < -1
1/2 if -1 <= z < 1
1/2 + 1/4(z-1 if 1 <= z < 2
1 if 2 <= z

What is P(Y = 2)?

Find all the numbers t with the property that both P(Y <= t) >= 1/2 and P(Y >= t) >= 1/2

2. Relevant equations

3. The attempt at a solution

For the P(Y=2) I integrated at the point 2 plus and minus epsilon and came up with 1/4z - 0 where z =2. Thus, 1/2. My concern is that this problem has a lot of constants Thus, I would expect P(Y=x) to equal 0 everywhere but in [1,2). Then I have no idea how to find the median of the distribution function. Sorry, if these are easy questions. The class is being taught without a book and I'm afraid I'm not used to that.

2. Nov 16, 2007

### Hurkyl

Staff Emeritus
I don't understand what you're doing -- could you spell it out? The answer is certainly wrong.

Not just a lot of flat-lines in the graph, but it's mostly continuous! P(Y=a) can only be positive if the graph of the cumulative distribution function has a discontinuity. (right?)

Do you know how to find P(Y<=t)?

3. Nov 16, 2007

### MathematicalPhysicist

well, no expert here, but to find P(Y=2) shouldn't you integrate f(z) over (-infinity,2]?

4. Nov 16, 2007

### sprint

F(z) = P(y<= z)

what exactly is "y" here?

the CDF is supposed to be F(z) = P[Z <= z]

5. Nov 17, 2007

### Hurkyl

Staff Emeritus
He said that f was the cumulative distribution, not the probability density.

6. Nov 17, 2007

### HallsofIvy

Staff Emeritus
This is impossible. The integral of a probability density function over its entire domain must be 1. You obviously can't have "f(z)= 1 if 2<= z". It also cannot be a cumulative probabililty distribution which was my next guess. I have no idea what is intended here.

No, that would give P(Y$\le 2$). With a continuous probability density, the probobability that Y is any specific number is 0.