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October 23rd, 2015, 05:30 AM  #1 
Newbie Joined: Jan 2014 Posts: 10 Thanks: 0  Description of distribution w. indicator random variable
Dear Forum, I was discussing this very interesting problem with my friend, and I want to hear your opinions on it. Let $\displaystyle X$ be a random variable with unknown distribution, and $\displaystyle c$ be a userdefine parameter. Let $\displaystyle A$ be the event that $\displaystyle X = x \leq c $ Define the random variable $\displaystyle Y$, such that $\displaystyle Y = y = g $ if $\displaystyle A$ occurs and $\displaystyle Y = y = 0$ otherwise. How do i give a formal distribution of $\displaystyle Y$, i.e. expected value and variance? If the general case is too difficult, we can assume that $\displaystyle X$ is normally distributed 
October 23rd, 2015, 05:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,625 Thanks: 622 
Since Y has only two possible values (g or 0) it cannot be normally distributed. Since the distribution for X is unknown, you can't say much about the distribution for Y.

October 25th, 2015, 04:57 AM  #3 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  The OP was saying we can assume X, not Y, is normally distributed... in which case the distribution for X is known.

October 25th, 2015, 08:12 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,625 Thanks: 622 
Sorry to have misread the question. The distribution for Y, even if X is normal, will be determined by the mean and variance for X.

October 26th, 2015, 09:23 AM  #5 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics 
If $\displaystyle X \sim N(\mu, \sigma^2)$ and the cparameter is $\displaystyle \mu$, then $\displaystyle Y \sim Bernoulli(0.5)$ and thus $\displaystyle E(Y) = 0.5$ and $\displaystyle Var(Y) = 0.5(10.5) = 0.25$. Not exactly groundbreaking, though :P


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