Question

The joint density function for continuous random variable (X, Y) is given by

Find
(a). P[0<=X+Y<=1]
(b). Find marginal densities for X and Y respectively. Are X and Y independent random variables? Justify your answer.
(c).

Answer 1

Answer on Question #46538 – Math – Statistics and Probability

The joint density function for continuous random variable $(X, Y)$ is given by

$$f_{XY}(x, y) = \begin{cases} 6e^{-(2x+3y)}, & x \geq 0, y \geq 0 \\ 0, & \text{e.w} \end{cases}$$

Find

(a). $P[0 \leq X + Y \leq 1]$

(b). Find marginal densities for $X$ and $Y$ respectively. Are $X$ and $Y$ independent random variables? Justify your answer.

Solution

(a)

$$P[0 \leq X + Y \leq 1] = \int_{0}^{1} dx \int_{0}^{1-x} dy \cdot 6e^{-(2x+3y)} = \int_{0}^{1} dx \cdot 2e^{-2x} \int_{0}^{1-x} dy \cdot 3e^{-3y}.$$ $$\int_{0}^{1-x} dy \cdot 3e^{-3y} = \int_{0}^{1-x} d(3y) e^{-3y} = \int_{0}^{3(1-x)} dx e^{-z},$$

where $z = 3y$

$$\int_{0}^{3(1-x)} dx \, e^{-z} = -e^{-z} \Big|_{0}^{3(1-x)} = e^{0} - e^{-3(1-x)} = 1 - e^{3x-3}.$$ $$\begin{aligned} P[0 \leq X + Y \leq 1] &= \int_{0}^{1} dx \cdot 2e^{-2x} (1 - e^{3x-3}) = \int_{0}^{1} dx \cdot 2e^{-2x} - \int_{0}^{1} dx \cdot 2e^{x-3}. \\ &\quad \int_{0}^{1} dx \cdot 2e^{-2x} = e^{0} - e^{-2} = 1 - \frac{1}{e^{2}}. \end{aligned}$$ $$\int_{0}^{1} dx \cdot 2e^{x-3} = \frac{2}{e^{3}} \int_{0}^{1} dx \, e^{x} = \frac{2}{e^{3}} (e - 1) = \frac{2}{e^{2}} - \frac{2}{e^{3}}.$$ $$P[0 \leq X + Y \leq 1] = 1 - \frac{1}{e^{2}} - \left(\frac{2}{e^{2}} - \frac{2}{e^{3}}\right) = 1 - \frac{3}{e^{2}} + \frac{2}{e^{3}} = 0.69.$$

(b)

$$f_{X}(x) = \int_{0}^{\infty} dy \cdot 6e^{-(2x+3y)} = 2e^{-2x} \int_{0}^{\infty} dy \cdot 3e^{-(3y)} = 2e^{-2x} (-e^{-3y} \Big|_{0}^{\infty}) = 2e^{-2x} (1 - 0) = 2e^{-2x}.$$ $$f_{Y}(y) = \int_{0}^{\infty} dx \cdot 6e^{-(2x+3y)} = 3e^{-3y} \int_{0}^{\infty} dx \cdot 2e^{-(2x)} = 3e^{-3y} (-e^{-2x} \Big|_{0}^{\infty}) = 3e^{-3y} (1 - 0) = 3e^{-3y}.$$ $$f_{X}(x) \cdot f_{Y}(y) = 2e^{-2x} \cdot 3e^{-3y} = 6e^{-(2x+3y)} = f_{XY}(x, y).$$

Therefore $X$ and $Y$ are independent random variables.

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