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October 23rd, 2015, 05:30 AM   #1
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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 user-define 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
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October 23rd, 2015, 05:33 PM   #2
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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.
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October 25th, 2015, 04:57 AM   #3
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Quote:
Originally Posted by mathman View Post
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.
The OP was saying we can assume X, not Y, is normally distributed... in which case the distribution for X is known.
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October 25th, 2015, 08:12 PM   #4
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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.
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October 26th, 2015, 09:23 AM   #5
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If $\displaystyle X \sim N(\mu, \sigma^2)$ and the c-parameter is $\displaystyle \mu$, then $\displaystyle Y \sim Bernoulli(0.5)$ and thus $\displaystyle E(Y) = 0.5$ and $\displaystyle Var(Y) = 0.5(1-0.5) = 0.25$. Not exactly groundbreaking, though :P
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