Distribution of a function of a Uniform distributed variable

[lucky999]

If R = 300000 - 500000U, and if U ~ Uniform[0,1], then how do we work out the parameters of the distribution of R??

I know it is uniform, but I seem to have had a mind block!
I mean if we take E(R) and Var(R), then we can have an "approximate" normal distribution.
But how do we actually work it out....I know I need to use the tables on page 13...:S

Thanks in advance

 

[John Lee]

If R = 300000 - 500000U, and if U ~ Uniform[0,1], then how do we work out the parameters of the distribution of R??

I know it is uniform, but I seem to have had a mind block!
I mean if we take E(R) and Var(R), then we can have an "approximate" normal distribution.
But how do we actually work it out....I know I need to use the tables on page 13...:S

Thanks in advance

By parameters do you mean the range?

In which case it's quite straightforward:

0 < U <1

So

300000 - 500000×0 > 300000 - 500000U > 300000 - 500000×1
300000 > R > -200000

note the signs swap as we've multiplied by a negative number.

 

[lucky999]

Many thanks for your reply John! That makes sense now.

But what about say:
R = 300000 - 500000P

where P ~ Poisson(5)

How would we find the distribution for R (I assume its Poisson?) and what would be the parameter?


And suppose R = 300000 - 500000B
where B ~ Binomial (100,0.25)

Again how would we find the distribution of R?

 

[Calum]

There is no general algorithm that you can follow to get the distribution of f(X) given the distribution of X. There are various techniques you can use:

http://www.statlect.com/subon2/dstfun1.htm
http://www.math.montana.edu/~jobo/st421/chap6n.pdf

but I have to admit my approach more often than not is to fire up a copy of R, create a simulation, and have a look at the histogram of the results.

 

[suraj]

But what about say:
R = 300000 - 500000P

where P ~ Poisson(5)

How would we find the distribution for R (I assume its Poisson?) and what would be the parameter?

First of all, I am using

R = 3 - 5P for convenience

Now, if
P = 0 , 1 , 2 , 3 , 4 ...... then
R = 3 , -2 , -7 , -12 , -17 ......

So R is taking Poisson probabilities but for different set of values.
I am not sure that we can say R is Poisson because it is taking negative values, but we can certainly find its PDF by starting like this:

Prob.(R = r)
= Prob.( 3 - 5P = r)
= Prob.(P = x)

where, x = (3 - r)/ 5

So its PDF is [ exp(-5) * 5^x / x! ]

with mean "-22" and variance "125"
and range of R is

r = 3 , -2 , -7 , -12 , -17 ......

Similarly you can do this for Binomial dist. also

 

[John Lee]

Many thanks for your reply John! That makes sense now.

But what about say:
R = 300000 - 500000P

where P ~ Poisson(5)

How would we find the distribution for R (I assume its Poisson?) and what would be the parameter?


And suppose R = 300000 - 500000B
where B ~ Binomial (100,0.25)

Again how would we find the distribution of R?

Answers above are good.

In general, for DRV you just take the same probabilities but apply them to different X's (if that makes sense).

Whereas for CRV you have to use the function of a RV (at the end of Ch3) to obtain the new f(x).