Final answer:
The provided joint probability density function is analyzed to determine the normalizing constant, calculate various probabilities, compute the expected value and variance of X, and find the marginal and conditional distributions for variables X and Y.
Step-by-step explanation:
To solve these problems, we need to use the properties of the joint probability density function (pdf) for continuous random variables X and Y. The density function provided is f(x, y) = c(x + y). The boundaries for x and y are given as 0 < x < 3 and x < y < x + 2.
Finding the constant c
To find the value for c that makes f(x, y) a valid pdf, we must ensure that the total area under the pdf equals 1. We calculate this by integrating the function over the given bounds:
\(\int_{0}^{3}\int_{x}^{x+2} c(x + y) dy dx = 1\)
Calculating Probabilities
For parts (b) through (e), (i), (j), and (k), we calculate the appropriate probabilities by integrating the density function over the relevant regions specified by the conditions given.
Expected Value and Variance of X
To find the expected value of X (E(X)) and variance of X (VAR(X)), we use the marginal probability density function of X, which we find by integrating the joint pdf over y for each fixed x (part h). Then we use these formulas:
E(X) = \(\int x f_X(x) dx\)
VAR(X) = E(X^2) - [E(X)]^2 = \(\int x^2 f_X(x) dx\) - [E(X)]^2
Marginal and Conditional Distributions
The marginal probability density function for X, f_X(x), is found by integrating the joint pdf over all possible values of y. The conditional probability density function for Y given that X = 1 is found by setting x = 1 in the joint pdf and scaling it so that its integral over y equals 1.