MHB Calculate Probability: Diff $Y-X \leq 1$

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To calculate the probability that the difference \( Y - X \leq 1 \) where \( X \) is uniformly distributed on \([0,2]\) and \( Y \) on \([2,4]\), the joint probability distribution \( f_{X,Y}(x,y) \) is used, defined as \( f_{X,Y}(x,y) = p_1(x)p_2(y) \). The relevant probability can be expressed as \( P(Y - X \leq 1) = \iint_{y-x \leq 1} f_{X,Y}(x,y) \, dx \, dy \). A graphical representation illustrates the areas where the joint density is non-zero and where \( Y - X \leq 1 \). The discussion emphasizes the use of the density functions for independent variables to find the desired probability. Understanding the joint distribution is crucial for solving this problem effectively.
evinda
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Hello! (Wave)

Suppose that $X$ has the uniform distribution on the interval $[0,2]$ and $Y$ has the uniform distribution on the interval $[2,4]$. If $X,Y$ are independent, I want to find the probability that the difference $Y-X$ is $\leq 1$.

I have thought the following.The density function of $X$ is

$$p_1(x)=\left\{\begin{matrix}
\frac{1}{2} &, 0 \leq x \leq 2 \\
0 & , \text{ otherwise}
\end{matrix}\right.$$

while the density function of $Y$ is

$$p_2(x)=\left\{\begin{matrix}
\frac{1}{2} &, 2 \leq x \leq 4 \\
0 & , \text{ otherwise}
\end{matrix}\right.$$How can we find the probability that the difference $Y-X$ is $\leq 1$ ? Do we use the above density functions? (Thinking)
 
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Hey evinda! (Smile)

Yep, we use those density functions.

$X$ and $Y$ have a so called joint probability distribution $f_{X,Y}(x,y)$.
If means that:
$$P(Y-X\le 1) = \iint_{y-x\le 1} f_{X,Y}(x,y)\,dx\,dy$$
And since they are independent, we have:
$$f_{X,Y}(x,y) = f_X(x)f_Y(y) = p_1(x)p_2(y)$$
(Thinking)
 
Just for fun, here's the corresponding graph.
[DESMOS]advanced: {"version":5,"graph":{"squareAxes":false,"viewport":{"xmin":-0.5,"ymin":-0.5,"xmax":6.5,"ymax":6.5}},"expressions":{"list":[{"id":"graph1","type":"expression","latex":"\\max\\left(\\left|x-1\\right|,\\left|y-3\\right|\\right)\\le1","color":"#60a0e3"},{"id":"2","type":"expression","latex":"y-x\\le1\\left\\{\\max\\left(\\left|x-1\\right|,\\left|y-3\\right|\\right)\\le1\\right\\}","color":"#c74440"},{"id":"3","type":"expression","color":"#fa7e19"}]}}[/DESMOS]

The blue area corresponds to the part where the joint probability density is non-zero.
And the red area corresponds to the part where $Y-X \le 1$. (Thinking)
 
Nice, thank you! (Smirk)
 
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