Relating probability distributions huh?

[Dafydd]

Homework Statement

Problem description:
A variable X has expected value 0.002 in meters. Consider X - 0.002, scale to millimeter, and we get Y.

Tasks:
a) Express Y as a function of X
b) Relate the probability distributions F_X and F_Y
c) Relate the probability density functions f_X and f_Y

Homework Equations

$$F(x) = \operatorname P ( X \leq x ) = \int_{-\infty}^x f(t) \, \mathrm{d}t$$

$$\int_{-\infty}^{\infty} f(x) \, \mathrm{d}x = 1$$

$$f(x) = F'(x) \geq 0$$

The Attempt at a Solution

a) Y = 1000X - 2
(I think)

b) I have no clue. I mean, I could make a wild guess, but I don't see any reason to.

c) I suppose we get f_X and f_Y by differentiating F_X and F_Y... somehow.

I also don't really know what it means to "relate" these things. Is it to express the one in terms of the other, so that for example if a*b = c then relating a to b I either say that a = c/b or that b = c/a? Or what?

 

[LCKurtz]

Homework Statement

Problem description:
A variable X has expected value 0.002 in meters. Consider X - 0.002, scale to millimeter, and we get Y.

Tasks:
a) Express Y as a function of X
b) Relate the probability distributions F_X and F_Y
c) Relate the probability density functions f_X and f_Y

Homework Equations

$$F(x) = \operatorname P ( X \leq x ) = \int_{-\infty}^x f(t) \, \mathrm{d}t$$

$$\int_{-\infty}^{\infty} f(x) \, \mathrm{d}x = 1$$

$$f(x) = F'(x) \geq 0$$

The Attempt at a Solution

a) Y = 1000X - 2
(I think)

That looks good.

b) I have no clue. I mean, I could make a wild guess, but I don't see any reason to.

c) I suppose we get f_X and f_Y by differentiating F_X and F_Y... somehow.

Start by looking at the cumulative distribution function for Y:

$$G(y) = P(Y\le y) = P(1000 X - 2 \le y) = P(X \le\ ??)\ ...$$

You should be able to calculate this in terms of f_X and it's derivative with respect to y will give you the density function for y.