The amount of kerosene, in thousands of liters, in a tank at...

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 $$ f_{J}(x, y)=\left\{\begin{array}{lc} 2, & 0< x < 1 y, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Determine if $X$ and $Y$ are independent. (b) Find $\mathrm{P}(1 / 4< X < 1 / 2 \mid Y=3 / 4)$.

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
$$
f_{J}(x, y)=\left\{\begin{array}{lc}
2, & 0< x < 1 y, \\
0, & \text { elsewhere. }
\end{array}\right.
$$
(a) Determine if $X$ and $Y$ are independent.
(b) Find $\mathrm{P}(1 / 4< X < 1 / 2 \mid Y=3 / 4)$.

Key Concepts

Joint Density Function

A joint density function describes the probability distribution of two continuous random variables simultaneously. It specifies how probability is assigned over the range of possible pairs of values and must integrate to one over the entire support of the random variables.

Independence of Random Variables

Two random variables are independent if the joint density function factors into the product of their individual marginal densities. This means that knowing the value of one variable does not provide any additional information about the other.

Marginal Density Function

The marginal density function of a random variable is obtained by integrating the joint density function over the other variable's range. It provides the probability distribution of one variable irrespective of the other, summarizing its individual behavior.

Conditional Density Function

The conditional density function of one random variable given the value of another is derived by dividing the joint density by the marginal density of the given variable. It represents the probability distribution of one variable when the other variable is fixed at a specific value.

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event or condition is known to be true. In the context of continuous random variables, it involves integrating the conditional density function over an interval of interest.

Support of Random Variables

The support of random variables refers to the set of values for which the joint or marginal density is nonzero. It defines the region where the probability mass or density is concentrated and is essential for correctly setting up integrations in probability calculations.

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