# Random Variables

#### Julio

##### Member
Let $\Omega=\{\omega_1,\omega_2,\omega_3\}$ an sample space, $P(\omega_1)=P(\omega_2)=P(\omega_3)=\dfrac{1}{3},$ and define $X,Y$ and $Z$ random variables, such that

$X(\omega_1)=1, X(\omega_2)=2, X(\omega_3)=3$

$Y(\omega_1)=2, Y(\omega_2)=3, Y(\omega_3)=1$

$Z(\omega_1)=3, Z(\omega_2)=1, Z(\omega_3)=2.$

Show that these three random variables have the same probability distribution. Find the probability distribution of $X+Y.$

Hi !, my name is Julio. I'm from Chile, my english isn't good, sorry, but try to do the best I can. Some indication for the problem?, thanks!

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Let $\Omega=\{\omega_1,\omega_2,\omega_3\}$ an sample space, $P(\omega_1)=P(\omega_2)=P(\omega_3)=\dfrac{1}{3},$ and define $X,Y$ and $Z$ random variables, such that

$X(\omega_1)=1, X(\omega_2)=2, X(\omega_3)=3$

$Y(\omega_1)=2, Y(\omega_2)=3, Y(\omega_3)=1$

$Z(\omega_1)=3, Z(\omega_2)=1, Z(\omega_3)=2.$

Show that these three random variables have the same probability distribution. Find the probability distribution of $X+Y.$

Hi !, my name is Julio. I'm from Chile, my english isn't good, sorry, but try to do the best I can. Some indication for the problem?, thanks!
Hi Julio! Welcome to MHB! Hint: What is the probability distribution of X?
That is, what is the probability that X=1, X=2, respectively X=3?

#### Julio

##### Member
Thanks I Like Sirena.

The probability distribution of an random variable $X$ is $P(X=x)=p(x)$ if $X$ is discrete, and $P(X=x)=f(x)$ if $X$ is continuous.

But really don't understand how solve the problem, help me please ! #### Klaas van Aarsen

##### MHB Seeker
Staff member
Thanks I Like Sirena.

The probability distribution of an random variable is $P(X=x)=p(x)$ if $X$ is discrete, and $P(X=x)=f(x)$ if $X$ is continuous.

But really don't understand how solve the problem, help me please ! So the question is what is $P(X=1)$?
Since $P(\omega_1)=\frac 1 3$ and $X(\omega_1)=1$, it follows that $P(X=1) = \frac 1 3$.
Similarly $P(X=2)=P(X=3)=\frac 1 3$.

How about Y? Does it have the same distribution?

#### Julio

##### Member
OK, thanks!

The query is how know that $\omega_1=1$?, I don't is notation used.

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#### Klaas van Aarsen

##### MHB Seeker
Staff member
OK, thanks!

The query is how know that $\omega_1=1$?, I don't is notation used.
You don't.
What you do know, is that $\omega_1$ is a possible outcome with probability $1/3$.
And furthermore, if the outcome is $\omega_1$, that $X = 1$.

#### Julio

##### Member
Thanks, then

$P(Y=1)=P(Y=2)=P(Y=3)=P(\omega_n)=\dfrac{1}{3}.$

Truth?

Then, the random variables have the same probability distribution because all are equal $\dfrac{1}{3}$?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Thanks, then

$P(Y=1)=P(Y=2)=P(Y=3)=P(\omega_n)=\dfrac{1}{3}.$

Truth?

Then, the random variables have the same probability distribution because all are equal $\dfrac{1}{3}$?
Yep! #### Julio

##### Member
One question, what I find strange is that these random variables take these different values​​, and in the end, your probabilities give the same result. #### Klaas van Aarsen

##### MHB Seeker
Staff member
One question, what I find strange is that these random variables take these different values​​, and in the end, your probabilities give the same result. Ah. But we still have the second part of the question, which sort of addresses that.

#### Julio

##### Member
Ah. But we still have the second part of the question, which sort of addresses that.
Oh truth !, I had forgotten. Good, I think that is only add. Namely,

$X(\omega_1)+X(\omega_2)+X(\omega_3)=1+2+3=6,$

$Y(\omega_1)+Y(\omega_2)+Y(\omega_3)=2+3+1=6.$

Thus, $(X+Y)(\omega_n)=6+6=12.$

Okay?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Oh truth !, I had forgotten. Good, I think that is only add. Namely,

$X(\omega_1)+X(\omega_2)+X(\omega_3)=1+2+3=6,$

$Y(\omega_1)+Y(\omega_2)+Y(\omega_3)=2+3+1=6.$

Thus, $(X+Y)(\omega_n)=6+6=12.$

Okay?
I'm afraid not.
The first possible outcome is $\omega_1$ with probability $\frac 1 3$.
With $X(\omega_1)=1$ and $Y(\omega_1)=2$, that means that $X+Y=1+2=3$.
So $P(X+Y=3) = \frac 1 3$.

The interesting part is that even though X and Y have the same probability distribution, we get to be surprised by the probability distribution of the sum X+Y.