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I am trying to figure out this question.

Two statistically dependent random variables $x_1$ and $x_2$ are applied at the inputs to a threshold detector, the output from which is equal to the number of inputs that exceed the threshold $T$. Thus, $y = 0; 1$ or $2$.

Determine the density function $p_Y (y)$ in terms of $p_{X_1 X_2}(x_1; x_2)$ and $T$.

From this I know:

$X_1 = \{ 0$ if $x_1< T , 1 $o.w$\}$, $X_2 = \{0$ if $x_2< T , 1$ o.w$\}$ and $Y = \{ 0$ if $x_1$ and $x_2 < T$ , $1$ if $x_1$ or $x_2 < T$, $2$ o.w$\}$.

How do I calculate the joint probability $p_Y(y)$?

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  • $\begingroup$ Your title says "Two statistically independent input" while your question says "Two statistically dependent random variables" $\endgroup$
    – Henry
    Jan 26, 2016 at 21:41
  • $\begingroup$ Thanks, I fixed the title. $\endgroup$ Jan 30, 2016 at 23:46

1 Answer 1

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Assuming all the random variables are non negative, and given $Y$ is a discrete random variable, you have

  • $\displaystyle \mathbb{P}(Y=0)=\int_{x_1=0}^T \int_{x_2=0}^T p(x_1, x_2) \,dx_2\,dx_1$

  • $\displaystyle \mathbb{P}(Y=1)=\int_{x_1=T}^\infty \int_{x_2=0}^T p(x_1, x_2) \,dx_2\,dx_1 + \int_{x_1=0}^T \int_{x_2=T}^\infty p(x_1, x_2) \,dx_2\,dx_1$

  • $\displaystyle \mathbb{P}(Y=2)=\int_{x_1=T}^\infty \int_{x_2=T}^\infty p(x_1, x_2) \,dx_2\,dx_1 $

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