What is Probability distribution: Definition and 201 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.
I am doing a study of the possibility of transition between 12 different events. I have a dataframe with these key events (listed from 1 to 12) over a period of time. I constructed a transition probability matrix between these events (photo of the matrix is attached below). As I don't have a...
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...
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...
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?
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Assume a double-slit like experiment, but instead of double...
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...
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...
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:
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...
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...
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...
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...
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...
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...
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...
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.
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 !
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...
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)...
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...
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...
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...
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...
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...
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...
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...
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...
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##...
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)...
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##...
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...
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...
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 $$Pr(M|\lambda) =...
Hi,
I am doing a past paper but I am kinda stuck on one of the questions.
These are the answers I have:
2a. 225/260 = 0.8654
2b. 32/260 * 4/32 = 0.01407
2c. 32/260 * 28/32 + 228/260 * 221/228 = 0.9577
Then for 2d, I have no idea what to do. Am i suppose to draw one of those probability...
Hey! :o
Let $M$ be a measure space and $(a_i)_{i\in \mathbb{N}}\subset M$. I want to show that for positive $p_1, \ldots , p_n$ with $\displaystyle{\sum_{i=1}^np_i=1}$ by $\displaystyle{Q=\sum_{i=1}^np_i\delta_{a_i}}$ a probability distribution is defined. Do we have to show that...
Since it's a PDF, that means the entire area under the curve must be 1, so
$\displaystyle \begin{align*} \int_0^1{ a \left( x^2 + b \right) \,\mathrm{d}x } &= 1 \\ a \left[ \frac{x^3}{3} + b\,x \right] _0^1 &= 1 \\ a \left[ \left( \frac{1^3}{3} + b\cdot 1 \right) - \left( \frac{0^3}{3} + b...
Consider the Gaussian position measurement operators $$\hat{A}_y = \int_{-\infty}^{\infty}ae^{\frac{-(x-y)^2}{2c^2}}|x \rangle \langle x|dx$$ where ##|x \rangle## are position eigenstates. I can show that this satisfies the required property of measurement operators...
I have some data that I want to do simple linear regression on.
However I don't have a lot of datapoints, and would like to know the uncertainty in the parameters. I.e. the slope and the intercept of the linear regression model.
I know it should be possible to get a prob. distribution of the...
Hello I am trying to check if the Method of Moments and Maximum Likelihood Estimators for parameter $\theta$ from a sample with population density $$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...
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...
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...
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...
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...
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...
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...
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...
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
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
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...
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...