Can you post the exact problem (I'm assuming this is a paraphrasing).
I think the title says it all but I'll elaborate a little...
We want to know the variance of a random variable X which is:
- equal to zero 80% of the time
- equal to a random variable that is ~ Uniform(1,b) the other 20% of the time.
I have searched all over the internet for this answer and have come up donuts. Your help and even just insights would be greatly appreciated. Thanks!
The question in the OP is a problem I need to solve for a project (not a problem out of a textbook) so there's no "problem" to refer to. I'll try to describe the question in the OP a little differently:
Suppose X: S --> [0, 10] is a continuous random var defined on sample space S with pdf =
Pr(X = 0) = 0.8
Pr(X) = (1-p(0))/(b - a) = 0.2/10 for all X within (0, 10]
Find Var(X).
In other words, for X = 0, the pdf of X is given by a constant (namely 0) and for X within (0,10], the pdf of X is given by Uniform(0,10].
I realize that I can simply use the Var(X) = E[(X-u)^2] formula to solve this but I'd like to make sure there's no analytical solution before I go that route.
Worded more generally, the question is as follows: Is there an analytical solution to find the variance of a r.v. X with different pdfs for different values of X.
Thanks!