Statistics help

[HJ123]

Hey guys I've got several questions about statistics.

Here's the first one.

1. Suppose X is a discrete random variable that takes on the three values x1, x2, x3 with probabilities p1, p2, p3 respectively. Describe how you could generate a random sample from X if all you had access to were a list of numbers generated at random from the interval [0, 1].

3. Suppose X is a continuous random variable with c.d.f. FX(x) = P(X  x). To make things easier, suppose further that X takes on values in an interval [a, b] (do allow for a to be −1 and b to be +1).

Let Y be the random variable defined by Y = FX(X). This looks strange, but is perfectly
valid since FX is just a function, and you are allowed to take functions of random variables.

Your problem: show that Y distributed U[0, 1].

Method: calculate the c.d.f. Y , FY (y) = P(Y  y), for all real y. Replace Y with FX(X),
and consider when you can take the inverse of FX.
Taking the inverse of FX is not always possible, in particular, for x < a and x > b. Consider
those cases separately.
You will find that the c.d.f. of Y is 0 for y < 0, y for 0  y  1, and 1 for y > 1, so indeed
Y distributed U[0, 1].
An important application of this fact is that if u is a random selection from the interval [0, 1], F−1 X (u) is a random selection from X.

 

[jolly]

:?

 

[HJ123]

So basically no one knows how to do this question??

 

[stapel]

So basically no one knows how to do this question?

Or maybe the volunteers who are proficient in statistics are out of town on vacation...?

It is currently the American Thanksgiving holiday weekend; you may have to wait until next week for people to get back to work, so they can surf by during lunch or whatever.

Eliz.

 

[pka]

I have not idea what #1 is about!

But #3 is standard.
Because \(\displaystyle Y = F_X\) the range of Y is \(\displaystyle [0,1]\).

So that if \(\displaystyle 0 \le a < b \le 1\; \Rightarrow \;\left( {\exists c} \right)\left[ {F_X (c) = a} \right]{\rm{ and }}\left( {\exists d} \right)\left[ {F_X (d) = b} \right]\).

Moreover, if \(\displaystyle a \le Y \le b\; \Rightarrow \;F_X (d) - F_X (c) = b - a\).

Thus \(\displaystyle P\left( {a \le Y \le b} \right) = \frac{{b - a}}{{1 - 0}}\).

That shows that Y is uniform on [0,1].

 

[tkhunny]

So basically no one knows how to do this question??

More accurately, no one cares to wade through all that. You dumped half your text book. (q.v. hyperbole) If you truly do not understand anything about that problem, nothing we can do here will help you. PKA got you started. Let's see where you are going.

1) Maybe chop up the interval [0,1] into chunks of size p1, p2, and p3?