Let X and Y denote the lengths of life, in years, of two...

Let $X$ and $Y$ denote the lengths of life, in years, of two components in an electronic system. If the joint density function of these variables is
$$
f(x, y)=\left\{\begin{array}{ll}
e^{-(x+y)}, & x>0, y>0, \\
0, & \text { elsewhere }
\end{array}\right.
$$
find $P(1<X<2 \mid Y=2)$.

Key Concepts

Integration in Continuous Distributions

Integration is the primary tool for calculating probabilities in continuous distributions. Since probabilities are given by the area under the curve of the density function, definite integrals over specified intervals are used to determine the likelihood that a random variable falls within those intervals.

Conditional Probability Density Function

The conditional probability density function defines the distribution of one continuous random variable given that another is fixed. It is calculated by dividing the joint density by the marginal density of the conditioning variable, and it is useful for finding probabilities or expectations when one variable is known to take on a specific value.

Independence of Random Variables

Two random variables are independent if their joint density function can be factored into the product of their individual marginal densities. This concept simplifies the analysis because the distribution of one variable is unaffected by conditioning on the other, allowing for straightforward computation of conditional probabilities.

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood that two continuous random variables take on a pair of values. It provides the foundation for determining probabilities by integrating over regions in the two-dimensional space of the variables.

Marginal Probability Density Function

The marginal density of a random variable is derived by integrating the joint density over the range of the other variable. It represents the probability distribution of one variable irrespective of the value of the other, and is essential for understanding the behavior of individual variables within a joint distribution.

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