Probablity Question - Joint PDF Expectation/Variance

• MCooltA
In summary, the student attempted to solve a homework problem but was unclear about the notation and ended up getting incorrect results.
MCooltA

Homework Statement

I have been given a joint PDF for X and Y, with ranges Y>X≥0.

I need to find the E(x) and E(y).

Homework Equations

I know E(x) = ∫(x)*(f(x,y) dx and E(y) = ∫(y)*(f(x,y)) dy

The Attempt at a Solution

For ∫x*f(x,y) dx, i used the limits = x to ∞

For ∫y*f(x,y) dy, i used the limits = 0 to y

Is this correct?

MCooltA said:

Homework Statement

I have been given a joint PDF for X and Y, with ranges Y>X≥0.

I need to find the E(x) and E(y).

Homework Equations

I know E(x) = ∫(x)*(f(x,y) dx and E(y) = ∫(y)*(f(x,y)) dy

The Attempt at a Solution

For ∫x*f(x,y) dx, i used the limits = x to ∞

For ∫y*f(x,y) dy, i used the limits = 0 to y

Is this correct?

These are both wrong. EX = int x*f(x,y) dx dy, etc.

RGV

Im confused by what you meant above, but i meant to say;

I have found f(x) by ∫f(x,y) dy, with the limits x to ∞.

To then find the E(x) do i ∫x * f(x) dx, with the limits 0 to y?

MCooltA said:
Im confused by what you meant above, but i meant to say;

I have found f(x) by ∫f(x,y) dy, with the limits x to ∞.

To then find the E(x) do i ∫x * f(x) dx, with the limits 0 to y?

OK, this is equivalent to what I wrote. You really should use different symbols for the different distributions, such as g(x) = int f(x,y) dy and h(y) = int f(x,y) dx, or use f_X(x) instead of g(x) and f_Y(y) instead of h(y).

As to your second question: the region in (x,y) space is {0 <= x <= y}, so yes, for any given x, y goes from x to infinity. However, once y has been "integrated out" it is no longer present, so NO, in EX = int x*f_X(x) dx, x does NOT go from 0 to y---there is no y now!. The variable x goes from 0 to infinity: when x was 5, y went from 5 to infinity, when x was 10 million, y went from 10 million to infinity, etc.

RGV

What is a joint probability density function (PDF)?

A joint probability density function (PDF) is a function that describes the probability of multiple random variables taking on specific values simultaneously. It is used to calculate the probability of events that involve multiple variables or outcomes.

How is the expectation of a joint PDF calculated?

The expectation of a joint PDF is calculated by taking the sum of the product of each possible value of the random variables and their corresponding probabilities. This can also be represented as an integral for continuous random variables.

What is the relationship between joint PDF and marginal PDF?

The joint PDF describes the probability of multiple variables occurring together, while the marginal PDF describes the probability of a single variable occurring alone. The marginal PDF can be calculated by integrating the joint PDF over all possible values of the other variables.

What is the role of covariance in joint PDF?

Covariance measures the relationship between two random variables. In the context of a joint PDF, it is used to determine if the two variables are independent or if there is a relationship between them. A covariance of 0 indicates independence, while a non-zero covariance indicates a relationship.

How do you calculate the variance of a joint PDF?

The variance of a joint PDF can be calculated using the definition of variance, which involves taking the difference between each possible value of the random variables and the expected value, squaring them, and weighting them by their corresponding probabilities. This can also be represented as an integral for continuous random variables.

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