Consider the following joint probability density function of...

Consider the following joint probability density function of the random variables $X$ and $Y:$ $$ f\{x, y)=\left\{\begin{array}{ll} \frac{3 x-y}{9}, & 1 < x < 3,1< y<2, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Find the marginal density functions of $X$ and $Y$. (b) Are $X$ and $Y$ independent? (c) Find $P(X>2)$.

Consider the following joint probability density function of the random variables $X$ and $Y:$
$$
f\{x, y)=\left\{\begin{array}{ll}
\frac{3 x-y}{9}, & 1 < x < 3,1< y<2, \\
0, & \text { elsewhere. }
\end{array}\right.
$$
(a) Find the marginal density functions of $X$ and $Y$.
(b) Are $X$ and $Y$ independent?
(c) Find $P(X>2)$.

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Key Concepts

Joint_Probability_Density_Function

A joint probability density function (joint pdf) describes the probability distribution of two continuous random variables simultaneously, specifying the likelihood of the pair of outcomes occurring together. It is defined over a specified region and must satisfy the property that its integral over the entire space equals one.

Marginal_Density_Function

A marginal density function is obtained by integrating the joint probability density function over the range of one of the variables, which effectively 'marginalizes out' that variable. This function gives the probability distribution of a single random variable independent of the other variable.

Independence_of_Random_Variables

Two random variables are independent if the joint probability density function can be expressed as the product of the individual marginal density functions. Independence implies that the occurrence of one event does not affect the probability of the other.

Probability_Computation

Probability computation in the context of continuous distributions often involves integrating the density function over the desired range. This technique is applied to calculate the likelihood of events defined by inequalities, such as finding the probability that a random variable exceeds a certain value.

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