Uniform distribution- probabilities

[Dassinia]

Hello, I am stuck at this exercise:

1. Homework Statement
X ~ U(0, a), a > 0 and Y = min(X; a=2).
- Find the cumulative distribution function of Y
-Is the variable Y continuous ?

Homework Equations

3. The Attempt at a Solution [/B]
The density function for X is
f(t)= 1/a if 0≤t≤a , 0 elsewhere
Is it correct to write that :
∀t < 0, P(Y ≤ t) = 0
∀0 ≤ t < a/2, P(Y ≤ t) = P(X ≤ t) = t/a
∀t ≥ a/2, P(Y ≤ t) = 1

And then don't know what to do
Thanks

 

[haruspex]

I'm not familiar with the notation min(X; a=2). What does it mean?

 

[Ray Vickson]

Hello, I am stuck at this exercise:

1. Homework Statement
X ~ U(0, a), a > 0 and Y = min(X; a=2).
- Find the cumulative distribution function of Y
-Is the variable Y continuous ?

Homework Equations

3. The Attempt at a Solution [/B]
The density function for X is
f(t)= 1/a if 0≤t≤a , 0 elsewhere
Is it correct to write that :
∀t < 0, P(Y ≤ t) = 0
∀0 ≤ t < a/2, P(Y ≤ t) = P(X ≤ t) = t/a
∀t ≥ a/2, P(Y ≤ t) = 1

And then don't know what to do
Thanks

You have an '$a$' in the definition of $X$ itself, and an '$a$' in the "definition" of $Y$ in terms of $X$. Are they supposed to be the same '$a$' in both places? If so, I cannot make any sense out of what you wrote.

On the other hand, if you really mean that $Y = \min(X,2)$, then that would have meaning. In that case it is important to distinguish between the two cases $0 < a \leq 2$ and $a > 2$.

 

[Dassinia]

Oh sorry, I didn't notice that there is a mistake in the expression of Y

It is Y=min(X; a/2)

 

[haruspex]

Oh sorry, I didn't notice that there is a mistake in the expression of Y

It is Y=min(X; a/2)

In that case your CDF for Y is correct. Draw it. Is it continuous?

 

[Dassinia]

No, it is not continuous !
But how to find the cumulative distribution function ?
Thanks !

 

[haruspex]

No, it is not continuous !
But how to find the cumulative distribution function ?
Thanks !

The CDF is what you wrote in the OP. You specified P(Y<=t) for all three ranges of t.

 

[Dassinia]

Oh, right ! I don't know why I thought that I had to find something else
Thanks for your answers !