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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  1. A

    A Constructing PDFs for Max Likelihood Density Estimation Problem

    I have the following constrained optimization problem corresponding to the maximum likelihood density estimation: $$ \begin{aligned} &\text{maximize} && L(f) \\ &\text{subject to} && f \in H \\ &&& \int_a^b f(x) \mathop{}\!\mathrm{d} x = 1 \\ &&& f(x) \geq 0...
  2. chwala

    Solve this problem that involves a probability density function

    I am refreshing on this; ..after a long time... Note that i do not have the solution to this problem. I will start with part (a). ##f(u)= 3u-\dfrac{3u^2}{2k}## with limits ##0≤u≤k## it follows that, ##3k - \dfrac{3k}{2}=1## ##\dfrac{3k}{2}=1## ##k=\dfrac {2}{3}## For part (b)...
  3. chwala

    Solve the problem involving probability density function

    This is the question: This is the ms solution- from Further Maths paper. My question is referenced to the highlighted part. I can see they substituted for the lower limit i.e ##x=1## to get: ##F(x)=\dfrac{x^3-1}{63}## supposing our limits were; ##2≤x≤4## would the same approach apply? Anything...
  4. A

    Probability/Random variables question

    Hello all, I am wondering if my approach is coreect for the following probability question? I believe the joint PDF would be 1 given that the point is chosen from the unit square. To me, this question can be reduced down to finding the area of 1/4 of a circle with radius 1. Any help is appreciated!
  5. chwala

    Problem involving Probability density function

    I just want to be certain, i think the inequality indicated is not correct...ought to be less than. Kindly confirm...This is a textbook literature.
  6. A

    A Solving an Integral involving a probability density function

    In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows: "For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...
  7. A

    I The probability density function for the double-slit experiment

    I am desperate. I've scoured the web for the formula for the probability density function for the interference pattern obtained in the double slit experiment with both slits open. So I want to know the probability density function and not the intensity function. I prefer not to have references...
  8. PainterGuy

    Random variable and probability density function

    Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...
  9. N

    A Probability Density Function: Converting Experimental Observations to PDF

    Hi All I am currently doing Master in data science. I came across the function PDF probability density function which is used to find cumulative probability(range) of a continuous random variable. The PDF probability density function is plotted against probability density in y-axis and...
  10. redtree

    I Separation of variables for Named Probability Density Distributions

    Given a probability density distribution ##P(\vec{x})##, for what named distributions is the following true: \begin{equation} \begin{split} P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n) \end{split} \end{equation}
  11. M

    Finding marginal distribution of 2d of probability density function

    Hi, I have question about finding marginal distributions from 2d marginal pdfs that lead to the probabilities being greater than 1. Question: If we have the joint probability distribution ## f(x, y) = k \text{ for} |x| \leq 0.5 , |y| \leq 0.5 ## and 0 otherwise. I have tried to define a square...
  12. iVenky

    I What would the pdf look like for 'N' chess computers with the same rating

    If we have 'N' advanced computers (where N-> infinity) each with exactly the same rating to begin with and make them play with each other for an infinite number of games, what would the shape of the probability density function of the ratings eventually look like?
  13. A

    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...
  14. A

    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...
  15. F

    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...
  16. megf

    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...
  17. Vicol

    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...
  18. shahbaznihal

    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...
  19. S

    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...
  20. Another

    I Please help explain the probability density function

    ##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...
  21. S

    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: $\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...
  22. B

    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.
  23. P

    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)...
  24. E

    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...
  25. 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)...
  26. T

    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...
  27. M

    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...
  28. T

    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...
  29. Saracen Rue

    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...
  30. Saracen Rue

    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...
  31. Saracen Rue

    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...
  32. NatFex

    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...
  33. Schwarzschild90

    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...
  34. Jan Wo

    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...
  35. T

    Integrating a theta function/2d probability density function

    Homework Statement A two-dimensional circular region of radius a has a gas of particles with uniform density all traveling at the same speed but with random directions. The wall of the chamber is suddenly taken away and the probability density of the gas cloud subsequently satisfies $$...
  36. BobblyHats97

    If X ∼ Uniform(−1, 1) find the pdf of Y = |X|

    This question is killing me. I know the graph is non-monotonic so i have to split up finding F(Y) for -1<Y and Y<1 but then what do I do with the modulus? >.< Any help would be greatly appreciated! Thank you so much x
  37. J

    How can I find the CDF and PDF of Y?

    Problem Let X be a uniform(0,1) random variable, and let Y=e^−X. Find the CDF of Y. Find the PDF of Y. Find EY. Relevant Equations http://puu.sh/kAVJ8/0f2b1e7b22.png My attempt at a solution If I solve for the range of y I get (1, 1/e), but because Y is not an increasing function, my...
  38. S

    Probability density function of simple Mass-Spring system.

    Homework Statement We know that after long run of simple mass-spring system, there should be a probability of finding the mass at certain points between -A and A.. Obviously in probability of finding the particle near A or -A is higher than finding the particle at 0, because the speed is the...
  39. SSGD

    Industrial Event Takt Times (production cycle times) and Probability

    Background: I am a Mechanical Engineer working as an Industrial Engineer. I have collected some data that is the amount of time that an event took to complete. I first assumed it would be normally distributed, but after plotting a histogram and a normal distribution with the data, I doubt...
  40. D

    Sketch and mathematically represent pdf of given signal

    Homework Statement I've been asked to sketch and mathematically represent the pdf of 2 signals: (a) rep2T{5 rect(t/T) - rect((t-T)/T)} (b) rep2T{2 rect(t/T) + 4Arect(2t/T)} A(t) is the lambda function A(t) = 1- t for 0 <= t <= 1 1+t for -1 <= t <= 0 Any help would be much...
  41. diracdelta

    Probability density function of uniform distribution

    Homework Statement Random variable X is uniformly distributed on interval [0,1]: f(x)=\begin{cases} 1 & \text{ if } 0\leq x\leq 1\\ 0 & \text{ else} \end{cases} a) Find probability density function ρ(y) of random variable Y=\sqrt{X} +1 I tried like this. Is it good, if no why not...
  42. T

    Probability density function of another function

    Hi guys, I'm working through a problem right now and would like to pick your brains on some stuff. I have an function: $$ f(r,\phi)= -\frac{1}{3} -cos(2\phi)(\frac{a^2}{r^2}) \hspace{0.5cm} for \hspace{0.5cm} a<r<b $$. I'm working in radial coordinates so r is the distance from a center and...
  43. V

    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...
  44. Y

    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...
  45. N

    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...
  46. C

    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 ## ?
  47. M

    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...
  48. A

    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?