Probability density Definition and 285 Threads

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. O

    Klein Gordon equation, probability density

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  2. S

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  3. 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...
  4. C

    Probability density function help

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  5. K

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

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  6. 1

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  7. T

    How to Calculate Probability Density Functions for Exact Numbers

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  8. S

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  9. S

    Quantum tunneling probability density

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  10. I

    Unification through probability density gradients

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  11. N

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  13. N

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  14. C

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  15. R

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  16. A

    Moment Generating Functions and Probability Density Functions

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  17. A

    Square of wave function gives us the probability density

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  18. M

    Probability Density Function Help

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  19. R

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  20. J

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  21. K

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  22. V

    Probability Density Function - Need Help

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  23. D

    Probability Density Function of two Resistors in Parallel

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  24. 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?
  25. S

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

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  26. C

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  27. T

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  28. J

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  29. C

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  30. C

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  31. A

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

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  32. J

    Probability Density in Quantum Mechanics

    Consider the wave function corresponding to a free particle in one dimension. Construct the probability density and graph it as a function of position. Is this wavefunction normalizable? Now, I think that the function should be Psi = C1*exp(ikx-iEt). Thus, the probability density should be...
  33. G

    Probability density of a particle

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  34. L

    How Does Joint Probability Density Determine Dependence Between Variables?

    If the values of the joint probability density of Y1 and Y2 are as shown: 0 1 2 total 0 1/12 1/6 1/24 35/120 1 1/4 1/4 1/40 63/120 2 1/8 1/20 ... 21/120 3 1/120 ... ... 1/20 ttl 56/120 56/120 8/120 1 whew ;-) ok Find a) P (Y1=0) b) P(Y2=1 | Y1=1) c P(Y2=1) e Check if Y1...
  35. frankR

    Quantum Physics: Probability density

    For the ground state of the hydrogen atom, evaluate the probabilty density psi^2(r) and the radial probability density of P(r) for the positions. a) r = 0 b) r = rb I confused how this probability function is used. What's the technique here? Thanks
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