Question

if x be a random variable and a is a constant ,then e [x- a] is minimum as a is​

Answer 1

Answer:

Step-by-step explanation:

Let X be a random variable and I now assume for simplicity that it is uniformly distributed on [0,1].

Now fix t∈[0,1] and let Yt be the random variable Yt:=min{t,X}.

I have problems computing for example the density

Computing the cdf is easy: Of course P(Yt≤y)=1−P(t>y,X>y). Now I tried to split this apart as if t was just random variable independet of X and this gives P(Yt≤y)=y if y<t and 1 if y≥t.

However, I feel that taking the derivative is not the appropriate her since Yt attains the value t with positive probability. So Yt seems to be a random variable that is neither a discrete nor a continuous random variable. So maybe a density does not make sense.

However, in order to be able to compute expectation and so on, I would need a density. In fact I guess everything becomes easy if I knew on which space Yt actually lived. However, I have problems formalizing that. Can anybody help.

Maybe for the uniform distribution everything is easy by direct arguments. But I would like to see a general approach which is valid for X having an continuous distribution.