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Your question is a bit unclear. I hope this answers your question, but if not, ask it again in a more specific manner.

The pdf of a continuous uniform random variable (X) distributed over the range a to b is f(x) = 1/(b-a).

If we consider x and y are dimensions, and each have a uniform distribution, then we can similarly define f(y) = 1/(c - d). Now, since these are independent, the joint distribution is f(x,y) = 1/(c-d) * 1/(b-a).

So, if we want to find the probability of X < x* and Y < y*, where x* is in the range of a to b and y* is in the range of c to d, we would integrate twice over f(x,y):

F(x*,y*) = integral from a to x* , integral from c to y* f(x,y) dy dx

Q: What is the probability density function for a uniforn two-dimentional random variable?

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A probability density function assigns a probability value for each point in the domain of the random variable. The probability distribution assigns the same probability to subsets of that domain.

The formula, if any, depends on the probability distribution function for the variable. In the case of a discrete variable, X, this defines the probability that X = x. For a continuous variable, the probability density function is a continuous function, f(x), such that Pr(a < X < b) is the area under the function f, between a and b (or the definite integral or f, with respect to x, between a and b.

The area under the pdf between two values is the probability that the random variable lies between those two values.

No. f is a letter of the Roman alphabet. It cannot be a probability density function.

The integral of the density function from the given point upwards.

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A probability density function assigns a probability value for each point in the domain of the random variable. The probability distribution assigns the same probability to subsets of that domain.

No. The probability that a continuous random variable takes a specific value is always zero.

A probability density function can be plotted for a single random variable.

The formula, if any, depends on the probability distribution function for the variable. In the case of a discrete variable, X, this defines the probability that X = x. For a continuous variable, the probability density function is a continuous function, f(x), such that Pr(a < X < b) is the area under the function f, between a and b (or the definite integral or f, with respect to x, between a and b.

The probability density function of a random variable can be either chosen from a group of widely used probability density functions (e.g.: normal, uniform, exponential), based on theoretical arguments, or estimated from the data (if you are observing data generated by a specific density function). More material on density functions can be found by following the links below.

The area under the pdf between two values is the probability that the random variable lies between those two values.

A probability density function (pdf) for a continuous random variable (RV), is a function that describes the probability that the RV random variable will fall within a range of values. The probability of the RV falling between two values is the integral of the relevant PDF. The normal or Gaussian distribution is one of the most common distributions in probability theory. Whatever the underlying distribution of a RV, the average of a set of independent observations for that RV will by approximately Gaussian.

No. f is a letter of the Roman alphabet. It cannot be a probability density function.

A random variable is a variable that can take different values according to a process, at least part of which is random.For a discrete random variable (RV), a probability distribution is a function that assigns, to each value of the RV, the probability that the RV takes that value.The probability of a continuous RV taking any specificvalue is always 0 and the distribution is a density function such that the probability of the RV taking a value between x and y is the area under the distribution function between x and y.

It will not. For the interval (x, x+dx) it may well give a non-zero probability. With a continuous distribution, the probability of any particular value is always 0. What the probability density function gives is the probability that the variable is NEAR the selected value.

The integral of the density function from the given point upwards.

probability density distribution