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Let $Ω = {0, 1, 2, 3}$, and, $P(\{k\}) = \frac{1}{4}$ for $k = 0, 1, 2, 3$.

Define two random variables $X (ω) = \sin \frac{πω}{2}$ and $Y (ω) = \cos \frac{πω}{2}$.

Find distributions and distribution functions of the random variables X and Y.

Compute P({ω ∈ Ω : X(ω) = Y(ω)}).

The range of $X$ and $Y$ are $\{-1,0,1\}$ and $\{-1,0,1\}$ respectively.

Thus $X$ and $Y$ are discrete random variables. We can specify the distribution of $X$ and $Y$ using the probability mass functions.

  • PMF of $X$

The mapping $X:\Omega\to\mathbb{R}$ is given by

  • $0\mapsto\sin 0=0 $
  • $1\mapsto\sin(\pi/2)=1$
  • $2\mapsto\sin(\pi)=0$
  • $3\mapsto\sin(3\pi/2)=-1$.

We can now find the PMF of $X$ as follows.

For any $x\in\mathbb{R}$, we have $\mathbb{P}(X=x)=\mathbb{P}(\{\omega\in\Omega:X(w)=x\})$. Thus $\mathbb{P}(X=-1)=\mathbb{P}(\{3\})=\dfrac{1}{4}$ $\mathbb{P}(X=0)=\mathbb{P}(\{0,2\})=\dfrac{2}{4}$ $\mathbb{P}(X=1)=\mathbb{P}(\{1\})=\dfrac{1}{4}$

In other words, the distribution, PMF, of $X$ is a function $f_X:\mathbb{R}\to\mathbb{R}$ given by $$f_X(x)=\begin{cases}\dfrac{1}{4} &\quad\text{ if }x=-1,1\\ \dfrac{2}{4} &\quad\text{ if }x=0 \\ 0 &\quad\text{ otherwise}\\\end{cases}$$

  • PMF of $Y$

The distribution, PMF, of $Y$ is a function $f_Y:\mathbb{R}\to\mathbb{R}$ given by
$$f_Y(y)=\begin{cases}\dfrac{1}{4} &\quad\text{ if }y=-1,1\\ \dfrac{2}{4} &\quad\text{ if }y=0 \\ 0 &\quad\text{ otherwise}\\\end{cases}$$

  • To find $\mathbb{P}(X=Y)$

How to do this?

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    $\begingroup$ Try calculating the value of X and Y for each element of $\Omega$. $\endgroup$ – markseeto Oct 29 '16 at 4:16
  • $\begingroup$ can you change your title to be more specific? $\endgroup$ – hxd1011 Oct 29 '16 at 21:56
  • $\begingroup$ I have answered this question here : quora.com/…. I guess the same OP has posted the question here, back then I edited the details of the question to phrase it correctly, and the question details were exactly the same as here. $\endgroup$ – Supreeth Narasimhaswamy Feb 6 '17 at 13:04

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