1. ## Continous Uniform Max[T,2]

Hi, how am I suppose to interpret max[T,2]?

So T is a continuous uniform RV on (1,5) and X=max[T,2]

But [T,2] is in between (1,5) correct?

so if we have T, we know that is has to be a value less than 2, right? and if we have a T>2 , then it is just T? That does not make sense to me.

I understand that T<= 2 is a CDF , and T>2 is a SF (both being w.r.t. to continuous uniform distribution, as stated).

But can anyone help me with visualizing max[T,2]?

Moreover, how would we interpret X=min[T,2]?

TY

2. ## Re: Continous Uniform Max[T,2]

$T$ is uniform on $[1,5]$

$X = \max(T,2)$

First thing to do is get the distribution of $X$

$P[X < x] = \left \{\begin{array}{ll}0 &x<2 \\\dfrac 1 4 &x=2 \\ \dfrac{x-1}{4} &2 < x \leq 5 \end{array}\right.$

differentiating we get

$p_X(x) = \dfrac 1 4 \delta(x-2) + \dfrac 1 4,~2 \leq x \leq 5$

taking expectations we get

$E[X] = 2 \dfrac 1 4 + \displaystyle \int \limits_2^5 \dfrac x 4~dx = \dfrac{25}{8}$

$E[X^2] = 4\dfrac 1 4 +\displaystyle \int \limits_2^5 \dfrac {x^2}{4}~dx = \dfrac{43}{4}$

$Var[X] = \dfrac{43}{4} - \left(\dfrac{25}{8}\right)^2 = \dfrac{63}{64}$

I'm not 100% on this. Another pair of eyes would be helpful.

3. ## Re: Continous Uniform Max[T,2]

I am still having a hard time deciphering what the max[T,2] is or how we come to that conclusion.

4. ## Re: Continous Uniform Max[T,2]

Could I see it as:

a<b<c<d :

1<t<2<5

5. ## Re: Continous Uniform Max[T,2]

This was their solution, so it is basically the same except they didn't really explain the distribution of X

6. ## Re: Continous Uniform Max[T,2]

Originally Posted by math951
$\max[T,2] = \begin{cases}2 &T<2 \\ T &2 \leq T \end{cases}$
in the case of $T$ being restricted the way it is this translates to
$\max[T,2] = \begin{cases}2 &T \in [1,2)\\ T &T \in [2,5] \end{cases}$