What is Probability distribution: Definition and 200 Discussions

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc.

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  1. C

    I Help with probability problem: Probability that one random Gaussian event will happen before another one

    For concretness I'll use atoms and photons but this problem is actually just about probabilities. There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom...
  2. NeophyteinPhysics

    The increase of Fidelity after non-selective measurement

    The above question is adopted from the exercise of Preskill's quantum information lecture note My attempt: (a) From the condition, ## p(\theta)\propto \sin^{(2N-4)}\theta \cos\theta ##. Normalizing the probability distribution would give the answer. This is because the weight of the phase of...
  3. Z

    I Free particle probability distribution

    Abstract: If a laser shoots photons at a pinhole with a screen behind it, we get a circular non-interference pattern on the screen. Is this distribution Guassian, and if not, what would its wave function be? ===================== Assume a double-slit like experiment, but instead of double...
  4. chwala

    Understanding the concept of Probability distribution

    Consider the attachment below; How did they arrive at ##F_X (u) = \dfrac{u-a}{b-a}## ? I think there is a mistake on the inequality, probably its supposed to be ##a≤u<b## and that will mean; $$F_X (u) =\dfrac{1}{b-a} \int_a^u du= \dfrac{1}{b-a} ⋅(u-a)$$ as required. Your thoughts...then i...
  5. J

    A The Probability Distribution of a Bosonic Field when Emitted

    If a bosonic field is probabalistic, and if it can be emitted (suddenly coming into existence), what determines its probability distribution when it is emitted from a fermion? In other words, one thinks (or at least I think) of a fermion field as already being in existence and already having...
  6. WMDhamnekar

    A Coin flipping problem (Markov chain)

    b)Suppose that the coin flipped on Monday comes up heads. What is the probability that the coin flipped on Friday of the same week also comes up heads? My attempt to answer this question:
  7. evceteri

    Neutron scattering probability distribution

    Hi, I'm reading Chapter 2-II of of Duderstadt & Hamilton's "Nuclear Reactor analysis". In the section "Differential scattering cross sections with upscattering" it is discussed the situation in which neutrons suffers elastic scattering collisions in a hydrogen gas at finite temperature T and the...
  8. chwala

    Solve the probability distribution and expectation problem

    This is the problem; Find my working to solution below; find mark scheme solution below; I seek any other approach ( shorter way of doing it) will be appreciated...
  9. Lynch101

    I The Probability Distribution and 'Elements of Reality'

    The below comment by @vanhees71 is an interesting one and I would be interested in exploring its implications. I am inclined to think that we can draw certain inferences about nature based on how we interpret the probability function and what it tells us about the elements of reality of the...
  10. Joan Fernandez

    A Why is the MGF the Laplace transform?

    The Laplace transform gives information about the exponential components in a function, as well as oscillatory components. To do so there is a need for the complex plane (complex exponentials). I get why the MGF of a distribution is very useful (moment extraction and classification of the...
  11. L

    B Probability Distribution with a resettable count to win

    Hi hi, I was thinking about this, all of this starts playing a game, I'll show a simplification: We ca win several times. We have a count ##n##, where is the max number of rolls until you win, let's say we can win a ##m## amount. In every roll we can win ##m## with a probability of ##p##. If...
  12. kshitij

    What is the ideal number of guesses to maximize your marks in MCQs?

    To approach this, I first assumed the case when the students attempts all the remaining questions. Probability that they gain 4 marks for a guess = ##\frac 1 4## Probability that they lose 1 for a guess = ##\frac 3 4## Now let us say the number of correct guesses = ##r## Now we should have at...
  13. chwala

    Probability distribution for discrete data

    this is a textbook problem shared on a whattsap group by a colleague... i have no problem in finding the value of ##k=0.08##, i have a problem with part (ii) of the problem. I have attached the solution here; how did they arrive at the probability distribution of ##y##? attached below is...
  14. Addez123

    15 people flip coins, find the probability distribution

    Say I make it so that the 2 coin flips count as a single number 1,2,3,4 representing head-head, head-tails, tails-head, tails-tails. Then what do I do? I'm just lost as to how I would even approach this problem.
  15. I

    MHB Probability distribution: Estimate the size of the fox population

    In a park , 200 foxes are tagged. In 100 sighting, 14 were tagged. Estimate the size of the fox population ? This is how approached . 200 tagged : Population 14 tagged : 100 (200x100/14)= Estimated Population = 1,429 I wonder if this is right !
  16. B

    B How to combine 2 distributions with different sample sizes?

    I apologise in advance for what is a very basic question for someone with a maths degree (it was a long time ago!). I have 2 distributions that look something like this (but with much bigger samples), in the form of (probability,outcome). The outcome is literally just a number. Distribution 1...
  17. Schwann

    B Can PDF values be equal to zero at some given points?

    Suppose we have a function which looks like this: It seems like it meets criteria of probability density functions: this function is asymptotic to zero as x approaches infinity and also it is not negative. My question is: if at some points this function reaches zero (as I have shown above)...
  18. vvaibhav08

    A Probability distribution: exponential of a quartic

    $$\int_{-\infty}^{+\infty} exp (-[ax + bx^2]^2) dx$$ $$a\&b\in R$$
  19. Robert Webb

    I Could the probability distribution itself be quantised?

    Everything is quantised when you look at it close enough. What about quantum probability waves themselves? If the quantum multiverse interpretation were true, then each quantum decision leads to a splitting of the universe. But this isn't a binary choice, it's a probability distribution. For...
  20. koulbichok

    I Gaussian probability distribution of formation PBH

    Hello. If we consider PBH formation from collapse of large density perturbation in the early Universe, a mass PBH depends on density contrast as And δ must be larger then . Also we have β — an abundance of black holes, it's the ratio of the PBH energy density to the total energy density, this...
  21. malawi_glenn

    I "Inverse" probability distribution question

    Hi, I think I am stuck in my understanding of "inverse" probability distributions. This is a question I would like to have help understanding. I want to figure out the distribution of number of trials for a given fixed number of successes and given probability for success for Bernoulli trials...
  22. J

    I Probability distribution of angle of asteroid entry to the atmosphere

    Used to play with gravitational attraction simulations ages ago. One thing I noticed it was difficult to get a small object to collide with a bigger spherical one vertically and far more likely to hit at an angle far from 0. Has the math of this been worked out for asteroids entering the Earth's...
  23. NatanijelVasic

    I Fourier Transform of a Probability Distribution

    Hi all :oldbiggrin: Yesterday I was thinking about the central limit theorem, and in doing so, I reached a conclusion that I found surprising. It could just be that my arguments are wrong, but this was my process: 1. First, define a continuous probability distribution X. 2. Define a new...
  24. B

    I Function to find the probability distribution of a stock price

    Hi all. I'm trying to find a formula that will calculate the probability distribution of a stock price after X days, using the assumption that the price change follows a normal distribution. In the spreadsheet, you can see the simulation I've made of the probability distribution of the price of...
  25. WMDhamnekar

    MHB Joint probability distribution of functions of random variables

    If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation. Hint:The joint density function can be obtained by differentiating the...
  26. malawi_glenn

    I Probability distribution of random events

    Hi Imagine we have a lottery, with chance of winning 1 in 1000 (1/1000). I have made computer simulations in order to find confidence levels for winning. At 1000 bought lottery tickets, the confidence of winning is 64.1% and 2000 bought lottery tickets the confidence of winning is 87.1% By...
  27. renec112

    Probability distribution momentum for particle

    Homework Statement A particle with mass m is moving on the x-axis and is described by ## \psi_b = \sqrt{b} \cdot e^{-b |x|}## Find the probability distribution for the particles momentum Homework Equations ## \Phi (p)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \Psi(x,0) \cdot e^{-ipx} dx##...
  28. 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)...
  29. Mehmood_Yasir

    Pedestrian wait time pdf

    Homework Statement Pedestrian are arriving to a signal for crossing road with an arrival rate of ##\lambda## arrivals per minute. Whenever the first Pedestrian arrives at signal, he exactly waits for time ##T##, thus we say the first Pedestrian arrives at time ##0##. When time reaches ##T##...
  30. chwala

    Statistics: probability distribution problem

    Homework Statement two indepedent observations ##X_1## and ##X_2## are made up of the continuous random variable having the probability density function ## f(x)= 1/k##, and ## 0≤x≤k## find a. the cumulative distribution of ##X## b. Find the probability distribution of M, the...
  31. G

    I Rewriting of equality in conditional probability distribution

    I don't get $$\frac{P[x<X<x+dx|N=n]}{dx}=f_{X|N}(x|n)$$ Can someone derive why? I would believe that $$f_{X|N}(x|n)=\frac{f(x,N)}{p_n(N)}$$ but I don't get how that would be the same. And I don't get that $$\frac{P[x<X<x+dx|N=n]}{dx}=\frac{P[N=n|x<X<x+dx]}{P[N=n]}\frac{P[x<X<x+dx]}{dx}$$ Can...
  32. Danny Boy

    A Quantum measurement operators with Poisson distribution

    The following is a somewhat mathematical question, but I am interested in using the idea to define a set of quantum measurement operators defined as described in the answer to this post. Question: The Poisson Distribution ##Pr(M|\lambda)## is given by $$Pr(M|\lambda) =...
  33. S

    MHB Understanding Sample Proportions and the Binomial Distribution

    Hi, I am doing a past paper but I am kinda stuck on one of the questions. These are the answers I have: 2a. 225/260 = 0.8654 2b. 32/260 * 4/32 = 0.01407 2c. 32/260 * 28/32 + 228/260 * 221/228 = 0.9577 Then for 2d, I have no idea what to do. Am i suppose to draw one of those probability...
  34. M

    MHB Defining a Probability Distribution with Measure Spaces and Delta Functions

    Hey! :o Let $M$ be a measure space and $(a_i)_{i\in \mathbb{N}}\subset M$. I want to show that for positive $p_1, \ldots , p_n$ with $\displaystyle{\sum_{i=1}^np_i=1}$ by $\displaystyle{Q=\sum_{i=1}^np_i\delta_{a_i}}$ a probability distribution is defined. Do we have to show that...
  35. P

    MHB James' question about a continuous probability distribution

    Since it's a PDF, that means the entire area under the curve must be 1, so $\displaystyle \begin{align*} \int_0^1{ a \left( x^2 + b \right) \,\mathrm{d}x } &= 1 \\ a \left[ \frac{x^3}{3} + b\,x \right] _0^1 &= 1 \\ a \left[ \left( \frac{1^3}{3} + b\cdot 1 \right) - \left( \frac{0^3}{3} + b...
  36. J

    A Discrete measurement operator definition

    Consider the Gaussian position measurement operators $$\hat{A}_y = \int_{-\infty}^{\infty}ae^{\frac{-(x-y)^2}{2c^2}}|x \rangle \langle x|dx$$ where ##|x \rangle## are position eigenstates. I can show that this satisfies the required property of measurement operators...
  37. M

    I Linear regression and probability distribution

    I have some data that I want to do simple linear regression on. However I don't have a lot of datapoints, and would like to know the uncertainty in the parameters. I.e. the slope and the intercept of the linear regression model. I know it should be possible to get a prob. distribution of the...
  38. J

    I Checking for Biased/Consistency

    Hello I am trying to check if the Method of Moments and Maximum Likelihood Estimators for parameter $\theta$ from a sample with population density $$f(x;\theta) = \frac 2 \theta x e^{\frac {-x^2}{\theta}} $$ for $$x \geq 0$, $\theta > 0$$ with $\theta$ being unknown. Taking the first moment of...
  39. J

    MHB Probability Distribution Problem

    Suppose that cars pass a certain intersection at a rate of 30 miles per hour. What is the probability that during a three-minute interval, no cars will pass the intersection? I am really just wondering which distribution to use. I thought is should be Poisson because it is asking for events...
  40. Charlie313

    Does a probability distribution correctly describe entropy?

    The colloquial statistical mechanics explanation of entropy as if it is caused by probability is dissatisfying to me, in part because it allows highly organized (i.e. with a real potential for work) arrangements to appear as 'random fluctuations', though with very low probability. But as far as...
  41. R

    Marginal Probability Distribution

    Homework Statement Two components of a laptop computer have the following joint probability density function for their useful lifetimes X and Y (in years): f(xy)=xe^(−x(1+y)) 0 <= x <= y 0 otherwise Find the marginal probability density function of X, fX(x). Enter a formula below. Use * for...
  42. S

    Calculating Probability using the Poisson Distribution

    Homework Statement On average, 2 students per hour come into the class. What is the probability that the time between two consecutive arrivals is in the interval <10 minutes; 50 minutes>. Homework Equations p(k)=P(Y=k)=((lambda*t)k*(e-lambda*t)/k! The Attempt at a Solution I've tried using...
  43. thegreengineer

    Binomial distribution problem

    Right now I'm having a problem with a statistics problem. More specifically with a binomial distribution problem. The problem says: There is a family composed by 8 children. Calculate the probability that 3 of them are girls As far as I know, binomial distribution formula says...
  44. S

    A Discrete Multivariate Probability Distribution

    Homework Statement A fair coin has a ##1## painted upon one side and a ##2## painted upon the other side. The coin is tossed ##3## times. Write down a sample space for this experiment. Let ##X_1## be the sum of the numbers obtained on the first ##2## tosses and ##X_2## be the sum of the numbers...
  45. E

    MHB Discrete Probability Distribution

    Okay, my online class has posed another word problem and I cannot seem to understand this week's material or how to formulate a solution. Here it is: Imagine you are in a game show, a money give-away! There are 4 prizes hidden on a game board with 16 spaces. One prize is worth \$4000...
  46. kenyanchemist

    Probability distribution curve for an electron in 2s and 2p

    hi, so my lecturer decides to give me manic depression by sending me on a wild goose chase. what is the general form of a plot of Ψ, Ψr2 and r2Ψ versus r for both Ψ2s and Ψ2p orbital... am not even sure i said it right So far I have only gotten the Ψ2r2 versus r
  47. kenyanchemist

    I Probability distribution plots

    hi, so my lecturer decides to give me manic depression by sending me on a wild goose chase. what is the general form of a plot of Ψ, Ψr2 and r2Ψ versus r for both Ψ2s and Ψ2p orbital... am not even sure i said it right
  48. I

    I Random variable and probability

    Hi! I'm searching for guidance and help since I don't know how to solve this problem. Here it is: a) The two-dimensional random variable (ξ,η) is uniformly distributed over the square K={(x,y): 0≤x≤1 , 0≤y≤1} . Let ζ=√ξ2+η2 me the distance between the origo and the point (ξ,η) . Calculate the...
  49. Z

    What exactly is a general probability distribution?

    I'm watching a Stat. Mech. lecture and in it, the lecturer mentions a "general probability distribution" but doesn't explain what exactly that is, yet in the context of the video, everything necessary to understand is understood. On some cursory google searches I'm finding several hits for a...
  50. M

    Math: discrete probability distribution

    Homework Statement We ask a person to taste 18 biscuits , 8 made to butter ( the other 10 are made to margarine ) , and to identify 8 butter cookies . He does not know the exact number of butter cookies . As he sees no difference , he randomly selects those he claims to be butter . Y = the...
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