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

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. a. Find $E\left(Y_{1} | Y_{2}=y_{2}\right)$ b. Use the answer derived in part (a) to find $E\left(Y_{1}\right)$. (Compare this with the answer found in Exercise 5.77.)

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.
a. Find $E\left(Y_{1} | Y_{2}=y_{2}\right)$
b. Use the answer derived in part (a) to find $E\left(Y_{1}\right)$. (Compare this with the answer found in Exercise 5.77.)

Key Concepts

Joint Probability Density Function

A joint probability density function (pdf) specifies the probability distribution of two or more random variables simultaneously. It describes the probability with which a set of values occurs and is used to compute probabilities over specific regions in the multivariate space by integrating over those regions.

Marginal and Conditional Distributions

The marginal distribution of a random variable is obtained by integrating the joint pdf over the other variable(s), effectively 'summing out' the unwanted variables. The conditional distribution, on the other hand, gives the probability distribution of one random variable given a fixed value of another. These concepts are crucial when breaking down joint distributions into more manageable parts and for computing related measures such as the conditional expectation.

Conditional Expectation

The conditional expectation is the expected value of one random variable given the occurrence of another event or specific value of another random variable. It is calculated by integrating the product of the variable and its conditional density over the allowable values, providing insight into the variable’s behavior when additional information is known.

Law of Total Expectation

The law of total expectation, also known as the tower property, states that the overall expected value of a random variable can be computed by taking the expectation of its conditional expectation with respect to another variable. This fundamental result allows for the decomposition of complex expectations into simpler, conditional parts, facilitating easier computation when dealing with joint distributions.

Transcript

00:01 Okay, given the joint density function for y1, y2, which is equal to times 1 minus y2, y1 is greater than 0, greater or equal to 0, less or equal to y2, less or equal to 1, it is equal to 0.

00:24 Given this joint density function, we want to find the conditional expectation, y1 under y2 being some little y2.

00:37 Okay, to find this conditional expectation, we want to first find the conditional density, right? which is equal, which by definition is equal to the joint density divided by the marginal density for y2.

00:59 Okay, then our first step is to find this marginal density by the definition, the density function for y2 will be the integral on the range for y1 over the joint density function.

01:18 Okay, dy1, this omega y represents the range for y1.

01:25 Okay, by our restriction here, we know y1 actually goes from 0 to y2, six times 1 minus y2, dy1, which is equal to six times 1, y1 evaluated by 2 and 0, which is equal to six times y2.

01:54 Okay, well, y2 is greater or equal to 0 and less or equal to 1, it is equal to 0 else, because once y2 is not contained in this interval, then the density is 0.

02:09 So here we are going to integral some, 0 function, if we integrate some 0 function, we'll always get 0.

02:21 We'll have the expression for this guy.

02:24 And by the definition, the density function, the conditional density function will be equal to 6 over the marginal density, okay? well, y1 greater than 0 and less than y2, and we just define to be 0 for other case, right? this is by our definition, because we cannot directly get this 0 from the division of 2 ,0, because it is meaningless.

03:07 But as we're going to discuss some density functions, so we can just define to be 0.

03:15 Okay, so we can do some consolidation.

03:21 That means this joint density function will be 1 over y2 in this range.

03:28 Okay, so the conditional expectation, y1 under y2 being y2, will be equal to the density function, the integral for omega y1 again, y1 times this guy, y2 being y2, dy1.

03:55 Okay, then it will be equal.

03:58 Again, y1 will go from 0 to y2...

Please give Ace some feedback