# What is Probability density function: Definition and 128 Discussions

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.
The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

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15. ### A How to transform a probability density function?

I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...
16. ### A How to change the support of a probability density function?

Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]: $$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$ Then the beta distribution with support...
17. ### I Relation with Hessian and Log-likelihood

I would like to demonstrate the equation (1) below in the general form of the Log-likelihood : ##E\Big[\frac{\partial \mathcal{L}}{\partial \theta} \frac{\partial \mathcal{L}^{\prime}}{\partial \theta}\Big]=E\Big[\frac{-\partial^{2} \mathcal{L}}{\partial \theta \partial...
18. ### I Probability Density Function of the Product of Independent Variables

How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...
19. ### Probability density function in classical mechanics

Probability density function plays fundamental role in qunatum mechanics. I wanted to ask if there is any analogous density function in classical mechanics. Obviously if we solve Hamilton equations we get fully deterministic trajectory. But it should be possible to find function which shows...
20. ### I Phase space density function and Probability density function

I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...
21. ### B Expectation of probability density function

E(X) of probability density function f(x) is \int x f(x) dx E(X2) of probability density function f(x) is \int x^2 f(x) dx Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ? I tried to find E(5 + 10X) from pdf. I did two ways: 1. I found E(X) then using...

##E = \frac{1}{2}(kx^2+m \dot{x}^2)## ## \frac{2E - kx^2}{m}=\dot{x}^2## ##\frac{dx}{dt} = \sqrt{\frac{2E - kx^2}{m}}## or ## dt = \sqrt{\frac{m}{2E - kx^2}}dx ## ⇒##= \frac{1}{\sqrt{\frac{2E - kx^2}{m}}}dx## My Question please help me. 1. I know ##T = 2\pi\sqrt{\frac{m}{k}} .## but i don't...
23. ### MHB Given probability density function find its cumulative distribution function

Hi :) Here's my problem along with what I've done. Here is the problem: That is the p.d.f. of a random variable X. I have to find the cdf. I don't know which I should do so I tried it two ways. First: $\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...
24. ### MHB Probability Density Function of Average Value of Log-Normal Trials

X is a random variable that follows the Log-Normal probability density function. n indipendent trials are carried out. We want to know the probability density function of the random variable Y, that is defined as the average value of the “n” outcomes of the trials described above.
25. ### Finding Probability Density Functions for Independent Random Variables

Homework Statement Hello! I'm trying to understand how to solve the following type of problems. 1) Random variables x and y are independent and uniformly distributed on the interval [0; a]. Find probability density function of a random variable z=x-y. 2) Exponentially distributed (p=exp(-x)...
26. ### I Probability density of an exponential probability function

I have a model where the probability is spherically symmetric and follows an exponential law. Now I need the probability density function of this model. The problem is the singularity at the origin. How can I handle this? P(r) = ∫p(r) dr = exp(-μr) p(r) = dP(r)/(4πr²dr) One way I tried to...
27. M

### Probability density function for a given problem

https://www.quora.com/How-can-I-find-the-probability-density-function-of-a-continuous-random-variable-in-a-given-problem/answer/Maxime-Denis-2 How can I find the probability density function of a continuous random variable in a given problem? A mass m swings at the end of a rope (of length L)...
28. ### I Understanding the probability density function

Hi all This is not a homework question but something work related which I am having difficulty understanding which I was hoping someone from the community could help me with. I am trying to understand how to interpret & create the probability density function plot from a set of data. For...
29. ### Finding k in a probability density function

Homework Statement Let X, Y, and Z have the joint probability density function f(x,y,z) = kxy2z for 0 < x, y < 1, and 0 < z < 2 (it is defined to be 0 elsewhere). Find k. Homework Equations Not sure how to type this in bbcode but: Integrate f(x,y,z) = kxy2z over the ranges of x (zero to...
30. ### I What Does The Probability Density Function Tell You?

Hello All I was wondering if someone could help explain what the probability density function tells you. I am trying to learn about surveying and the PDF keeps cropping up and I do not fully understand it. For example I have:- measured a single angle 15 times calculated my Standard Deviation...
31. ### Probability Density Function problem

Homework Statement Presume the relation ##\frac{x}{x+y^2}-y=x## is defined over the domain ##[0,1]##. (a) Rearrange this relation for ##y## and define it as a function, ##f(x)##. (b) Function ##f(x)## is dilated by a factor of ##a## from the y-axis, transforming it into a probability density...
32. ### Probability Density Function problem

Homework Statement Q6. A function, ##f\left(x\right)=\frac{ax+1}{\left(ax-1\right)^3-\frac{a}{\left(x-1\right)^2-1}}##, can be defined as a PDf over the domain ##(0, 2)##. Express answers to 4 decimal places unless specified otherwise; (a) Find the value of ##a## given that ##f(x)## is a PDf...
33. ### Difficult Probability Density Function Question

Homework Statement A function, ##f\left(x\right)=\left|a^{\frac{\sin \left(x\right)}{\ln \left(ax\right)}}-\frac{x}{a}\right|##, intersects with another function, ##g\left(x\right)=\left|\frac{sin(ln(\sqrt{x}-\sqrt{a}))}{x^2-a^2}\right|##, at point ##Q(b,f(b))## and point ##R(c,f(c))##. A...
34. ### I Sum of Probability Density Function > 1?

I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
35. ### Calculating Standard Deviation for a Probability Density Function?

Homework Statement Homework Equations See below The Attempt at a Solution \begin{align} \begin{split} p(x) = C \ x \ exp(-x/ \lambda) \end{split} \end{align} If $p(x)$ is a probability density function on the interval $0 \textless x \textless + \infty$ , then it follows...
36. ### Getting electron density from probability density function

Hello Lastly I was thinking a lot about electron density definition. It is not intuitive for me and I'm looking for any mathematical tool that could explain it to me more. My friend told me about idea to derivate it from propability density function using Dirac delta distribution. I'd like to...

45. ### Electron One Split Energy Probability Density Function

Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=69371&d=1399142463 2. The attempt at a solution I am working on the last problem now. Here is what I have got so far. Basically I have converted the coordinate space wave function to a momentum space wave...
46. ### Finding the probability density function given the eigenfunction

Homework Statement I need to find the probability density function given the eigenfunction Homework Equations \psi=C\exp^({\frac{ipx}{\hbar}-\frac{x^2}{2a^2}}) The Attempt at a Solution I tried to square the function but that gave me a nasty integral that I could not solve. I...
47. ### Probability density function problem

Homework Statement Let the probability density function##f(x) = (3/4) \cdot (1-x^2)## if x is between -1 and 1, and let ##f(x)=0## otherwise. What is the probability of ##P(X \leq 0.8 | X>0.5)##? Homework Equations The Attempt at a Solution I assume I have to rewrite the p.d.f. into a joint...
48. ### How does the probability density function handle infinity in integrals?

http://gyazo.com/02812d5d8f1d07c72153c9f66740e147 I've dealt with integrals with infinity before. When considering the part x >= 1 , do I take the limit as if it's a very large number? i.e. ## \int_0^{\infty} x^{-2.5} \ dx = 2/3 ## ?
49. ### Generating a probability density function

I am trying to create a simple implementation of the Bayes decision rule with minimum error criterion and I am running into a problem. Specifically, if I have a data set consisting of a number of feature vectors stored in rows, how can I generate a probability density function from this data...
50. ### Probability density function

If g and f are two normalized probability density functions is it then true in general that the convolution of f and g is normalized too?