What is Density function: Definition and 192 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. K

    Joint probability density function

    Let X, Y, and Z have the joint probability density function f(x, y, z) = kx(y^2)z, for x>0, y<1, 0<z<2 find k \int_{0}^{2}\int_{- \infty}^{1}\int_{0}^{\infty}kxy^2z dx dy dz This integral should equal 1. Is my procedure correct so far? I don't manage to solve the integral...
  2. D

    How can I use this density function as a likelihood function?

    I am trying to make a function which is exponential for a while, and then turns gaussian: f(l,d) = \lambda e^{-\lambda d} , 0 < d < l and f(l,d) = (1-\int_0^l \lambda e^{-\lambda d} dd) \frac{1}{\sigma \sqrt{2 \pi}} e^{-(d-l)^2/(2\sigma^2)} , l < d < \infty (That is supposed to be...
  3. A

    What is the Median of a Density Function and How Can I Solve for It?

    Homework Statement The probability density function is defined as f(x) = (4/81)x(9-x^2) for 0 <= x <= 3 = 0 for every other value Find the median value of the density. Homework Equations The median of a function is integral from m to infinity of f(x)dx = 1/2 The...
  4. S

    About the joint density function

    Could anyone help to give an example where in the same proba space, x and y have each the density function, while the joint density function does not exist? Thanks in advance, Best regards
  5. P

    'Triangular Distributions' Probability Density Function

    (\Triangular" distributions.) Let X be a continuous random variable with prob- ability density function f(x). Suppose that all we know about f is that a </= X </= b, f(a) = f(b) = 0, and that there exists a value c between a and b where f is at a maxi- mum. A natural density function to...
  6. M

    Normal probability density function

    Homework Statement A production line is producing cans of soda where the volume of soda in each can produced can be thought of as (approximately) obeying a normal distribution with mean 500ml and standard deviation 0.5ml. What percentage of the cans will have more than 499ml in them...
  7. Z

    Probability Density Function of a Quadratic Equation

    HI Can anybody tell me how to calculate a PDF of y, where y is a function of x, such that y = a X*X + bX + C (i.e. a quadratic equation), and X follows the Normal Distribution X ~N(0, sigma) Help anybody? Thanks
  8. T

    Probability Density Function

    I feel embarassed for asking, but is there a fast way to calculate this without using integration by parts? \int 2e^(-2x)x^-1dx, 0 <= x < infinity There's supposed to be some kind of trick, right?
  9. L

    Probability Density Function with an exponential random variable

    The question is: if X is an exponential random variable with parameter \lambda = 1, compute the probability density function of the random variable Y defined by Y = \log X. I did F_Y(y) = P \{ Y \leq y \} = P \{\log X \leq y \} = P \{ X \leq e^y \} = \int_{0}^{e^y} \lambda e^{- \lambda x} dx =...
  10. R

    Bounded Probability Density Function

    Let the random variable X have the probability density function f(x). Suppose f(x) is continuous over its domain and Var[X] is bounded away from zero: 0 < c < Var[X]. Claim: f(x) is bounded over its domain. Is this claim true? I don't think a counterexample like X ~ ChiSq_1 applies...
  11. P

    Probability density function of digital filter

    given that x has an exponential density function ie p(x) = exp (-x) and x(n) & x(m) are statistically independent. Now y(n) = x(n-1)+x(n) what is the pdf (probability density function) of y(n)
  12. M

    Expected value from a density function

    Hey, I know how to find the expected value from the density function when it is in the form: (example) | y^2 -1<y<1 | fy =| | 0 elsewhere Ey = integral(upper limit 1, lower limit -1)[y*y^2 dy) but, what if the density function looks like this...
  13. M

    Expected value from a density function

    Hey, I know how to find the expected value from the density function when it is in the form: | fy =
  14. T

    Finding the Mean Number of Screams on a Roller Coaster Ride

    Homework Statement The density function for the number of times the riders scream on a roller coaster is given by ... f(x) = \frac{1}{3\pi}(1 - cos(2x)) if 0 \leq x \leq 3\pi and 0 otherwise. Find the mean number of screams over the course of the ride. The answer: 4.71238898038...
  15. V

    Probability density function of a pendulum displacement

    Hi, I need a verification for this question. Can some one help me? Question: A man enters the pendulum clock shop with large number of clocks and takes a photograph. He finds that most of the pendulums were at the turning points and only a few were captured crossing the mid point. Why is it...
  16. S

    Finding the Probability Density Function

    Homework Statement A dial indicator has a needle that is equally likely to come to rest at an angle between 0 and Pi. Consider the y-coordinate of the needle point (projection on the vertical axis). What is the probability density function (PDF) p(y)? Homework Equations I know the...
  17. K

    Probability density function homework

    Find a constant c such that f(x,y)=cx2 + e-y, -1<x<1, y>0, is a proper probability density function. My idea: f(y) 1 =∫ f(x,y) dx -1 So I have found f(y), now I set the following integral equal to 1 in order to solve for c: ∞ ∫ f(y) dy = 1 0 Integrating, I get something like...
  18. C

    Probability density function help

    ok iv have been stuck on this problem for like 30 mins it says "suppose x is a continuous random variable taking values between 0 and 2 and having the probability density function below." the graph below shows a triangle with the coordinates (0,1) (2,0) then it ask what is the Probably...
  19. K

    Why Is Only One Solution for c Valid in This Probability Density Function?

    Q: Given f(x) = cx + (c^2)(x^2), 0<x<1. What is c such that the above is a proper probability density function? Solution: 1 ∫ f(x) dx = 1 0 => 2(c^2) + 3c - 6 =0 => c= (-3 + sqrt57) / 4 or c= (-3 - sqrt57) / 4 => Answer: c= (-3 + sqrt57) / 4 (the second one rejected)...
  20. T

    Derivation of exponential density function for air

    Homework Statement If atmosphere can be treated as an isothermal ideal gas of constant mean molecular mass m, show that density drops exponentially with height - ρ= [ρ0]e^-z/h - where h is a constant Homework Equations ρ= [ρ0]exp^-z/h (derivation of ...) ρ=density ρ0=initial density at...
  21. 1

    Uniform probability density function question

    [b]1. Homework Statement A vendor at a market buys mushrooms from a wholesaler for $3 a pound, and sells them for $4 a pound. The daily demand (in pounds) from custumers for the vendor;s mushrooms is a random variable X with pdf f(x) = 1/40 if 0 (greater than) x (less than) 40 and 0...
  22. F

    Density Function for X-Y on [0,1]

    hi.. Homework Statement what's the density function for X-Y if X and Y are independent and continously distributed on [0,1]?
  23. E

    Hazard to Density to Cumulative Distribution Function

    ... ...deleted
  24. S

    How Do You Determine the Constant in a Piecewise Probability Density Function?

    if x is a continuous random variable from -1 to 1...how do you find c: f(x) = c + x , -1 < x < 0 c - x, 0 < x < 1 Do I integrate each one? Where do I go from there? Thanks!
  25. M

    How Do You Calculate Probability Using a Density Function?

    A continuous random variable X has the density function f(x)=x for 0<x<1 2-x for 1 _<x<2 0 elsewhere. a. Show that P(0<X<2)=1 B. Find P(X<1.2). Please see the attached file. Thank
  26. L

    Help with Power Spectral Density Function derivation

    Homework Statement Given a two-level atom with transition frequency { \omega }_{ ji } \equiv { \omega }_{a} and spontaneous decay rate \gamma, we are asked to find an expression for the "power spectral density function" S(\omega), in terms of \omega, {\omega}_{a}, and \gamma. 2. The...
  27. R

    Probability Density Function, prove it

    Homework Statement This is my 1st post here, so I will do my best. The following question is part of a number of probability density functions that I have to prove. Once I have the hang of this I should be good for the rest, here is the question: Prove that the following functions are...
  28. K

    Probability : joint density function of 3 Normal Distributions

    X1, X2, X3 are independent gaussian random variables. Y1 = X1+X2+X3 Y2 = X1-X2 Y3 = X2-X3 are given. What is the joint pdf of Y1,Y2 and Y3 ?
  29. M

    Probability Density Function Help

    Probability Density Function...Help The probabiltiy density function of the time to failure of an electric component in hours is f(x)=e^{(-x/3000)/3000} for x > 0 and f(x) = 0 for x \leq 0 determine the probability that a) A component last more than 1000 hours before failure I know how...
  30. R

    How Many Fragments Are Found Within 10 Kilometers of a Volcanic Eruption?

    Not really a homework question, but a problem I don't get nonetheless. The density of fragments lying x kilometers from the center of a volcanic eruption is given by: D(r) = 1/[sqrt(x) +2] fragments per square kilometer. To 3 decimal places, how many fragments will be found within 10...
  31. B

    How do I find the centroid of a solid with given volume and no density function?

    Hi, I am having trouble with the following question. Can someone help me out? a) Find the volume of the solid that lies above the cone \phi = \frac{\pi }{3} and below the sphere \rho = 4\cos \phi . b) Find the centroid of the solid in part (a). For the volume I got 10pi which I am...
  32. M

    The Two-Particle Density function

    I have a question regarding the two-particle density function, in particular its Fourier transform. I know that in a liquid or gas the function n_2(\mathbf{R}_1, \mathbf{R}_2) is the probability that two particles will be found at \mathbf{R}_1 and \mathbf{R}_2. But what is the significance of...
  33. quasar987

    What is the definition of a density function in cartesian coordinates?

    I want a rigorous description of the density function (in cartesian coordinates) \rho(x,y,z). I suggest that we define a function M(x,y,z,V'), where V' is a volume of any given shape centered on the point (x,y,z), giving the mass contained in that volume. Then define the density function as...
  34. K

    Calculating Earnings PDF with Job-Based Payment: Mean and Variance

    How do I calculate the PDF of someone's earning followed by their mean and variance? This is the question: Given density function f(x) = 2.5 if 0.1 < x < 0.5 0 otherwise The person is paid by the # of jobs they finish rather than by the hour. They get 10$/job. Calculate...
  35. I

    Conditional density function - please

    conditional density function - need help please! given a signal x, is a random variable which is expontential with a mean of 3. it is transmitted through an additive gaussian noise channel, where the gaussian noise has a mean of -2 and a variance of 3. the signal and noise are...
  36. V

    Probability Density Function - Need Help

    Probability Density Function -- Need Help! Hi, Can someone please check my work if i did the problem correctly? thanks in advance. Here is the problem: Find the PDF of W = X + Y when X and Y have the joint PDF fx,y (x,y) = 2 for 0<=x<=y<=1, and 0 otherwise. here is my solution...
  37. D

    Probability Density Function of two Resistors in Parallel

    I have a problem where there are two resistors in parallel and I need to find the equivalent resistance. R1 = X and R2 = Y, and X and Y are independent random variables, uniform over the range of 100-120. If R equivalent = Z = XY/X+Y, what is probability density function of Z?
  38. D

    Probability Density Function of two Resistors in Parallel

    I have a problem where there are two resistors in parallel and I need to find the equivalent resistance. R1 = X and R2 = Y, and X and Y are independent random variables, uniform over the range of 100-120. If R equivalent = Z = XY/X+Y, what is probability density function of Z?
  39. S

    What is the conditional probability of P(X > 0.2 | X < 0.6)?

    Hi Guys, I am having some trouble trying to solve a probability density function question. ...If the density function is: f(x) = 9x^3, 0 < x 1. What is the conditional probability of P(X > 0.2 | X <0.6) ?? Any help would be greatly appreciated :)
  40. B

    Integration to d density function

    I have a problem with an integration, namely: Int(from 0 to x) (1-F(x-t)) dF(x) and do not know how to calculate...:-(
  41. T

    Understanding the Probability Density Function of a Wave Function

    Have known a wave function, how get does distributing density function of the probability?
  42. A

    Math Help: P(40 ≤ X ≤ 50, 20 ≤ Y ≤ 25) & P(4(X-45)^2+100(Y-20)^2 ≤ 2)

    Suppose that X and Y are independent random variables, where X is normally distributed with mean 45 and standard deviation 0.5 and Y is normally distributed with mean 20 and standard deviation 0.1. (a) Find \ P(40 \leq X \leq 50, \ 20 \leq Y \leq 25). Ans. ~0.5 (b) Find \...
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