2. Let $(X, Y)$ be a continuous bivariate random variable having the joint probability density function
$$
f(x, y)=c x y, \quad 0 \leq x \leq y \leq 2
$$
for some real constant $c, f(x, y)=0$ elsewhere.
(e) Compute $\mu_{X}, \mu_{Y}, \sigma_{X}^{2}, \sigma_{Y}^{2}, \operatorname{Cov}(X, Y)$, and $\rho$.

Solution

Question

2. Let $(X, Y)$ be a continuous bivariate random variable having the joint probability density function
$$
f(x, y)=c x y, \quad 0 \leq x \leq y \leq 2
$$
for some real constant $c, f(x, y)=0$ elsewhere.
(e) Compute $\mu_{X}, \mu_{Y}, \sigma_{X}^{2}, \sigma_{Y}^{2}, \operatorname{Cov}(X, Y)$, and $\rho$.

Solution

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the mean of $X$, $\mu_X$, we integrate $x$ times the joint probability density function $f(x, y)$ over the entire range of $X$ and $Y$. ‖ ‖ step 2 ⋮ The integral for $\mu_X$ is $\int_0^2 \int_0^y x \cdot cxy \, dx \, dy$. Using the result from asksia-ll calculator step 1, we have $\mu_X = \frac{2c}{3}$. ‖ ‖ step 3 ⋮ To find the mean of $Y$, $\mu_Y$, we integrate $y$ times the joint probability density function $f(x, y)$ over the entire range of $X$ and $Y$. ‖ ‖ step 4 ⋮ The integral for $\mu_Y$ is $\int_0^2 \int_0^y y \cdot cxy \, dx \, dy$. Using the result from asksia-ll calculator step 3, we have $\mu_Y = \frac{8c}{15}$. ‖ ‖ step 5 ⋮ To find the variance of $X$, $\sigma_X^2$, we calculate $\int_0^2 \int_0^y (x - \mu_X)^2 \cdot cxy \, dx \, dy$. ‖ ‖ step 6 ⋮ Using the result from asksia-ll calculator step 4, we have $\sigma_X^2 = \frac{8}{225}c(256c^2 - 256c + 75) - \mu_X^2$. ‖ ‖ step 7 ⋮ To find the variance of $Y$, $\sigma_Y^2$, we calculate $\int_0^2 \int_0^y (y - \mu_Y)^2 \cdot cxy \, dx \, dy$. ‖ ‖ step 8 ⋮ Using the result from asksia-ll calculator step 5, we have $\sigma_Y^2 = \frac{128c^2(24576c^4 - 49152c^3 + 38176c^2 - 13600c + 1875)}{16875} - \mu_Y^2$. ‖ ‖ step 9 ⋮ To find the covariance of $X$ and $Y$, $\operatorname{Cov}(X, Y)$, we calculate $\int_0^2 \int_0^y (x - \mu_X)(y - \mu_Y) \cdot cxy \, dx \, dy$. ‖ ‖ step 10 ⋮ Using the result from asksia-ll calculator step 6, we have $\operatorname{Cov}(X, Y) = \frac{128c^2(96c^2 - 96c + 25)(256c^2 - 256c + 75)}{16875} - \mu_X \mu_Y$. ‖ ‖ step 11 ⋮ To find the correlation coefficient $\rho$, we use the formula $\rho = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}$. ‖ ‖ step 12 ⋮ Using the results from steps 6, 8, and 10, we calculate $\rho$. ‖ ∻[question number] Answer∻ ∻Key Concept∻ ⚹To compute the mean, variance, covariance, and correlation coefficient for a continuous bivariate random variable, we use the joint probability density function and integrate over the appropriate ranges.⚹ ∻Explanation∻ ⚹The mean is the expected value of the random variable, the variance measures the spread of the distribution, the covariance indicates the direction of the linear relationship between the variables, and the correlation coefficient measures the strength and direction of the linear relationship between the variables.⚹◊What is the formula to calculate the covariance of X and Y for the given bivariate random variable?⍭ Generate me a similar question◊

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