In Exercise 5.9 we determined that f(y1, y2)={ 6(1-y2), 0 ≤ y1≤ y2≤ 1 0, elsew

step 1 For exercise 77A, we have EY sub 1 equal to 3 fourths minus 2 plus three halves, which equals on

In Exercise 5.9 we determined that f(y1, y2)={ 6(1-y2), ...

In Exercise 5.9 we determined that $$f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} 6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ is a valid joint probability density function. Find $$\text { a. } E\left(Y_{1}\right) \text { and } E\left(Y_{2}\right)$$ $$\text { b. } V\left(Y_{1}\right) \text { and } V\left(Y_{2}\right)$$ $$\text { c. } E\left(Y_{1}-3 Y_{2}\right)$$

In Exercise 5.9 we determined that $$f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1 \\
0, & \text { elsewhere }
\end{array}\right.$$ is a valid joint probability density function. Find $$\text { a. } E\left(Y_{1}\right) \text { and } E\left(Y_{2}\right)$$ $$\text { b. } V\left(Y_{1}\right) \text { and } V\left(Y_{2}\right)$$ $$\text { c. } E\left(Y_{1}-3 Y_{2}\right)$$

Key Concepts

Joint Probability Density Function

A joint probability density function (pdf) defines the likelihood of a pair (or more) of continuous random variables taking on particular values simultaneously. It must be nonnegative everywhere and integrate to one over the entire space, and the domain of the pdf is often specified by inequalities.

Marginal Distribution

The marginal distribution of one random variable is obtained by integrating the joint pdf over the range of the other variable. This allows us to understand the behavior of a single variable independent of the other, which is critical when calculating univariate statistics.

Expected Value

The expected value (or mean) of a continuous random variable is a measure of its central tendency and is found by integrating the product of the variable and its pdf over the variable's range. For joint distributions, expectations can be evaluated using either the joint pdf directly or via the marginal pdf if available.

Variance

Variance quantifies the spread or dispersion of a random variable around its mean. It is computed as the expected value of the squared deviation from the mean. In practice, this involves integrating the square of the variable’s deviation from its mean times its pdf over its range.

Linearity of Expectation

The linearity of expectation is a fundamental property stating that the expected value of a linear combination of random variables is equal to the same linear combination of their expected values. This holds true regardless of whether the random variables are independent, making it a powerful tool in evaluating expressions such as E(Y1 - 3Y2).

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