Understanding probability density function

[cdot]

So I understand how for a continuous random variable the probability of an exact value of X is zero, but then what is the value of f(x) if it's not a probability? I thought it was a probability similar to how the pmf for a discrete random variable was a piece-wise function that gave the probability for various values of X. But it can't be a probability because the function f(x) DOES take on a value for every single value of x. You plug in an x and out pops an f(x). If f(x) is indeed a probability then doesn't this contradict the idea that the probability for any single value of x is zero. So what is f(x)?

 

[mathman]

The probability for a continuous random variable (X) is given by the distribution function. Specifically P(a<X<b) = F(b)-F(a), where F(x) is the distribution function. The density function f(x) is simply the derivative, f(x) = F'(x)

Note: this assumes F is well behaved (absolutely continuous).

 

[Stephen Tashi]

So I understand how for a continuous random variable the probability of an exact value of X is zero, but then what is the value of f(x) if it's not a probability? I

Visualize a rod of variable mass lying along the x-axis with it's left end at zero. Let F(X) be the total mass of the rod between x= 0 and x = X. The interpretation of F(X) is straightforward, but what is the meaning of f(X) = F'(X)? You have to accept the idea of an "instantaeous rate of change of mass" at the point x = X. The point x = X does not have a mass, but the mass in a small interval of length dX around it can be approximated by f(X) dX.

The probability density function f(X) of a continuous random variable has an analogous interpretation. It is the instantaneous rate of change of the cumulative probability function.

Often when you are trying to remember or derrive formulas in probability, you can cheat and think about f(X) as being "the probability that x = X" to remember the correct answer.

This way of incorrect thinking seems to work out more often in probability theory than in physics. In physics, if you have a instantaeous rate, you often have to keep the dX's in picture and your answer may have powers of the dX's and ratio's of them, some of which vanish and some of which produce the answer.

 

[cdot]

Thank you Stephen.

 

[blue_raver22]

Thank you for your question. Probability density function (PDF) is a mathematical function that describes the probability distribution of a continuous random variable. It is used to calculate the probability of a random variable falling within a certain range of values.

As you correctly mentioned, for a continuous random variable, the probability of an exact value is zero. This is because the probability of a continuous random variable falling on a specific point on the number line is infinitesimally small. Instead, the probability is calculated for a range of values.

The value of f(x) in a PDF does not represent a probability, but rather the probability density at a specific value of x. It is important to note that the area under the curve of a PDF must equal 1, which means that the total probability of all possible values of x is 1.

Think of it this way - the probability of rolling a specific number on a fair six-sided die is 1/6. However, if we were to roll the die an infinite number of times, the probability of getting that specific number would approach 0. Similarly, the probability of a continuous random variable falling on a specific point is 0, but the probability of it falling within a range of values can still be calculated using the PDF.

In summary, f(x) in a PDF represents the probability density at a specific value of x, not the probability itself. I hope this helps clarify your understanding of probability density function.