3.47 The amount of kerosene, in thousands of liters, in a...

3.47 The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount $Y$ from which a random amount $X$ is sold during that day. Suppose that the tank is not resupplied during the day so that $x \leq y,$ and assume that the joint density function of these variables is $$ f(x, y)=\left\{\begin{array}{ll} 2, & 0<x \leq y<1 \\ 0, & \text { elsewhere } \end{array}\right. $$ (a) Determine if $X$ and $Y$ are independent. (b) Find $P(1 / 4<X<1 / 2 \mid Y=3 / 4)$.

3.47 The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount $Y$ from which a random amount $X$ is sold during that day. Suppose that the tank is not resupplied during the day so that $x \leq y,$ and assume that the joint density function of these variables is
$$
f(x, y)=\left\{\begin{array}{ll}
2, & 0<x \leq y<1 \\
0, & \text { elsewhere }
\end{array}\right.
$$
(a) Determine if $X$ and $Y$ are independent.
(b) Find $P(1 / 4<X<1 / 2 \mid Y=3 / 4)$.

Key Concepts

Joint Density Function

A joint density function specifies the probability distribution for two or more continuous random variables simultaneously, assigning a density to each point in the space and allowing probabilities to be computed via integration over regions of the space.

Marginal Distribution

A marginal distribution is obtained by integrating the joint density function over the range of the other variable(s). It represents the distribution of a single variable irrespective of the values of the other variables.

Independence

Two random variables are independent if the joint density function factors into the product of their marginal densities for all values in their support. This means that the probability of any event involving both variables can be separated into independent probabilities.

Conditional Density Function

The conditional density function of one variable given the value of another is derived by dividing the joint density by the marginal density of the given variable. It expresses how the distribution of one variable changes when a particular value of the other variable is known.

Conditional Probability

Conditional probability for continuous random variables is computed by integrating the conditional density function over the desired range. This calculates the likelihood that the variable falls within a specified interval, given a particular condition on another variable.

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