F(x, y)

Definition

In statistics, f(x, y) represents a joint probability density function (pdf) of two random variables, x and y. This function describes the likelihood of two continuous random variables occurring simultaneously, providing insight into their relationship and the overall distribution of their combined outcomes. The values of f(x, y) are non-negative and integrate to one over the entire space of possible values for x and y.

5 Must Know Facts For Your Next Test

1. The joint probability density function f(x, y) can be used to find probabilities associated with specific ranges for both variables by integrating over those ranges.
2. To find the marginal pdfs from f(x, y), you need to integrate f(x, y) with respect to the other variable; for example, the marginal pdf of x is obtained by integrating f(x, y) dy.
3. If f(x, y) can be factored into the product of two individual pdfs, then x and y are considered independent random variables.
4. Graphically, the joint pdf f(x, y) can be represented as a surface in three-dimensional space where the height above any (x, y) point indicates the likelihood of those values occurring together.
5. The total probability must always equal one when integrating the joint pdf over its entire range: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dx \, dy = 1$$.

Review Questions

• How do you interpret the value of f(x, y) in terms of joint probabilities?
  • The value of f(x, y) indicates the relative likelihood of observing specific pairs of values (x, y) for two continuous random variables. It is not a direct probability but rather a density; hence higher values indicate more likely combinations compared to lower values. To find actual probabilities from this density function, you would integrate f(x, y) over a defined range for both x and y.
• What steps would you take to derive marginal probability density functions from a given joint pdf f(x, y)?
  • To derive the marginal probability density functions from a joint pdf f(x, y), you would perform integration. Specifically, to find the marginal pdf for x, integrate f(x, y) with respect to y across its entire range: $$f_X(x) = \int_{-\infty}^{\infty} f(x, y) \, dy$$. Similarly, for the marginal pdf for y, integrate with respect to x: $$f_Y(y) = \int_{-\infty}^{\infty} f(x, y) \, dx$$. This process isolates each variable’s distribution while accounting for its relationship with the other variable.
• Evaluate how independence between two random variables is determined using their joint pdf f(x,y).
  • To determine if two random variables x and y are independent using their joint pdf f(x,y), check if it can be expressed as the product of their marginal pdfs: $$f(x,y) = f_X(x) imes f_Y(y)$$. If this equality holds true for all values in their domains, then x and y are independent. This means that knowing the value of one variable does not provide any information about the value of the other variable, reflecting no dependency between their distributions.

Related terms

Marginal Probability Density Function: The probability density function of a single random variable derived from a joint probability density function by integrating out the other variable.

Conditional Probability: The probability of an event occurring given that another event has already occurred, often represented as f(x|y) or f(y|x).

Joint Distribution: A distribution that shows the probability of two or more random variables occurring at the same time, often represented as f(x, y) for two variables.