In the textbook from uni I had for probability and Math Stat (Wackerly), we (the textbook) defined various random variables with a definition that went something as follows:
"A random Variable Y is said to have a ________ probability distribution if and only if for parameter _______ and parameter _______ the density function of Y is: Insert complicated function here
This troubled me because it offered very little (usually zero) justification for why that function exists or its derivation.
Where do all these density functions come from? Did statisticians just try to come up with random (no pun intended) functions that integrated to 1 and then hope they had meaning and relevance to our world? (I don't actually think this is the case)
Are there any resources that contain the justification/derivation for the PDFs and CDFs we commonly use?
Are these explanations just too complicated for my feeble mind, and the authors are simply sparing me from inevitable suffering...
Hey, I'm a mathematics student.
As of how I currently understand it, the probility density function of a normal distributed random variable doesn't have an anti-derivative. Although I've seen this said in a couple of books I would like to know whether there is a proof for this statement as I haven't come into contact with one yet. Thus I would like to ask whether it can be proven and if so how, if not why not and why do we still treat it as a fact? Thanks alot in advance!
Hi,
I want to generate pdfs from caps and floors to gauge the likely path of interest rates over the next couple of years.
I’m looking for some reasonably penetrable reading material with practical examples. Many moons ago I majored in mathematics but safe to say I’m incredibly rusty.
TIA
Are these two terms interchangeable? The best I’ve been able to come up with to answer this question is this quote from the first line of the wikipedia article on PDFs:
“a probability density function (PDF), or density of a continuous random variable,...”
which seems to imply these are the same, but would appreciate confirmation.
I'm very poor at stats (I have a biology degree, so I can do tests, but don't really work with theory), and I can't find a way to google this.
Suppose I have a probability density function, which gives the proportion of people that have an illness. Because it's a bounded function I can choose the beta distribution as a way of representing this. So f(x)=Beta(a,b) where a and b are shape parameters.
Now, suppose I say that all the people are given a treatment, which reduces the proportion by an average of 20%, how do I calculate the resulting distribution, i.e. the new proportion of illness in the population?
I've tried the following:
probIllness = seq(0,1,0.01)
density = dbeta(probIllness,3,3)
plot(density~probIllness,type="l",xlab="Proportion infected",ylab="Density")
reduction = (1-0.2)
newDensity= rep(0,length(density))
# Loop over each
for(i in 1:length(probIllness)){
# Calculate the new illness proportion for this point
temp = probIllness[i] * reduction
# Find the nearest point to this one
point = which.min(abs(probIllness-temp))
# Enter the new probability in the nearest point
newDensity[point] = newDensity[point] + density[i]
}
# Renormalise the pdf (to have the same sum as y, presumably 100 here since dx = 0.01)
newDensity= newDensity/ sum(newDensity) * sum(density )
lines(newDensity~probIllness,type="l")
I'm reasonably certain there must be a "proper" way of doing this.
Any help appreciated, thanks!
Hi, I'm currently studying a bit of probability theory. I'm trying to solve this exercise but I don't know if my work is correct.
Let X be a random variable characterized by the following pdf:
f(x) = k|x|^(1/3)rect(x/2)
a)Determine the value of k in order to let f(x) be a valid pdf.
b)Determine the CDF of X
c)Calculate P({X<1/2}U{X>-1/2})
For the first step, I used the normalization property of the pdf and found k = 2/3. Is this first step correct?
Next, I know that the CDF of a random variable can be obtained by integrating the pdf (function of y) in the interval from -infinity to x. This is where I start having troubles. Should I integrate from -1 to x because of the rectangular function in the pdf?
Haven't tried the last step yet.
Any help will be appreciated. Thanks in advance.
https://imgur.com/gallery/VFk3G4A
The solution is for (a) only. I don't understand how
f(x) = (2x(2-|x|)) / 4(0.5 x 2 x 2) was derived. I also don't understand how finding f(x) for (a,b) confirms that f(x) = 0 for elsewhere.
I had a bit of a question about fitting probability density functions (specifically the Weibull Distribution) to a set of data. I have seen a number of examples where the distribution is directly overlaid on the histogram (that has not been normalized) and they both share the same axis titles and I can't fully justify why this is, specifically with the y-axis.
From my understanding, the y-value for the distribution is the probability density, which can be described as the rate that the probability is changing over the unit that x is (or the probability per x). The y-axis for the histogram (in the examples I have seen) is simply the number of times that a specific outcome has occurred, so I am not entirely sure how these two can be equated. Should the axis titles be different? I'm not entirely sure why the standard layout seems to be sharing the same axis title?
I've also been having some difficulty trying to find a very clear and simple way to explain what exactly the Weibull Distribution fit on a histogram of data (that has not been normalized) means for people who have no background in statistics, so if anyone could help with that as well it would be greatly appreciated!
[Solved] Hello, everybody. Is there a way to mark areas μ, μ−2σ, μ−σ, μ+σ, μ+2σ (μ - mean, σ - standard deviation) in a probability density function graph picture below? How can I select specific value interval and colour a corresponding chart area? The result should look similarly to normal distribution.
Solution.: Vertical error bars for corresponding values
https://preview.redd.it/ril37fptlco51.png?width=1650&format=png&auto=webp&s=d59bcfa739f75e97afaaa8fa765fb667a2384af4
Hey guys! I have been thinking about this for a while now and have made no progress I believe. If Y is an exponential random variable of rate 10 and X is another random variable independent from Y that is uniformly distributed on [0,Y], what is the density function f_X? What I thought was that, given y=Y, X is uniformly distributed on [0,y], so that the density f(x|y)=1/y. Is this correct? How would I then find f_X ? Thanks!
They said the probability of the Exact one value of x is equal 0 given they are continuous , but isnt the probability of x equal the height of the point on the probability density function, yes it maynot exactly correct, but it isnt 0 either . People say probability of each point is zero make me feel weird if x is on the x-axis and f(x) is on y- axis signify the probability, isnt at the point (x,y): y is the probability of the value x ??
Hi. I don't have a solid background in statistics so the concept of probability density functions in the statistics course I'm taking is new to me.
I need to derive the median of a continuous distribution with the following density function:
https://preview.redd.it/zw4oq1d5jot51.png?width=468&format=png&auto=webp&s=4867b396f3cd8f46886e67d85e65f4a023541272
I computed the CDF which is -2/x and I arrived with the answer of -4 but I don't think a negative median is possible. Hope someone can help clarify things. Thank you!
This series has given me such a clear understanding that I feel obliged to kindly ask you to continue
So suppose that we are given the marginal density function of X and Y respectively and we want to find their joint density function. Given that marginal density function of either variable is obtained by integrating the other one out, can we reverse this process to obtain the joint density function from marginals?
Offhand, I have kicked around a few ideas
- Model the points as a gaussian, bound regions within 3 standard deviations, discretize the pdf to assign a probability density to each voxel, integrate over overlapping regions
Issues: computationally expensive, memory, not easily implementable in some languages
- Using a method like normalized mutual information
Issues: I don't believe this is defined for 3d points from my understanding. Am I wrong?
What else might you try?
But I'm curious if there are more standardized ways to do this, perhaps with some premade functions in Matlab?
Hello All,
I am an Amateur at statistics so please bare with me.
I have collected some data from a population to create a standard deviation and mean. I am using this to predict future outcomes based on the normal distribution probability density function (which assumes 'normal symetrical skewness/kurtosis.
If say, for example: The mean is 15, the standard deviation is 3, and if X=16 the probability density function that X>x is .369. Is there any simple calculation to adjust the normal distribution probability density function to reflect skewness and kurtosis. Can I take .369 (normal conditions) and input it into some type of calculation with Skewness (2.6) and Kurtosis(1.5) to determine my actual probability?
I don't understand this assumption. It's not true in classical mechanics where the probability can abruptly drop to zero so where does this assumption come from in QM?
Hello Guys,
Please help me with this question about probability density function.
I have attached a link to the question and to my answer too.
I want someone to help verify whether my answer is correct. I am not pretty sure.
Pretty much the title. I want to learn hpw to do stats problems that ask to evaluate probability density functions (i.e. f(x) = 2x+t/a if x >=0, f(x) = 0 if x < 0, for example) finding their E and Var, Maximum Likelihood estimators and stuff.
Do you have any resources from where I can learn this topic?
Thank you!
Hey! Do you guys have any good resources for learning joint probability density functions? I can't find anything
Preface (with math stuffs so you can ignore)
A quick crash course on PDFs/CDFs:
PDF: probability that you will fall into a group is the height of the bar.
CDF: probability that you will fall to the left of the group is the height of the bar. (Yes, this is the sum of all bars to the left on the PDF)
So it has recently been hotly debated (in my group of friends anyways) regarding the methods to calculate probabilities for pulling helpers. After a night of suffering peer review, the methods have been finalized and confirmed via simulation. The probabilities for pulling a helper before the system smacks you with a pity follow a geometric distribution (I was right all along!!) and the probability you get smacked by the pity is like 45% for the event banner (since 80 for pity, 1% pull rate), and roughly 37% for the other banners. Note that if you're cool you might be interested that:
https://preview.redd.it/lo0t2xg0bcr31.png?width=393&format=png&auto=webp&s=fa57aa10841284f0a93a0347502cb15036e23459
But we're getting off tangent here. The PDF for when you want to pull for one helper is as follows:
Note this simulation ran 100k times instead of 10k like the others
The tail is rather startling, but that's your 37% chance of getting pitied! ¯\_(ツ)_/¯
You might also wonder why the heck the probabilities decrease (eg. probability of 25-50 pulls < probability of 0-25), this would seem to be counter-intuitive! Shouldn't the probabilities be uniform all the way through? This is not the case for one rather simple reason: your counter gets reset the instant you successfully pull. Yes, this implies that the further you go, the fewer unfortunate people there are that are still pulling.
EDIT: We now have a formula so you don't have to simulate!!
Let a = number of pulls before pity and
p = probability of success on any pull and
E_p = expectation for number of pulls
Thus it follows that:
https://preview.redd.it/7qay8pvf3gr31.png?width=980&format=png&auto=webp&s=c57b8c5488592145cf70925df6e9a7bd00a4de89
The simplified formula:
https://preview.redd.it/wjv81e5k3gr31.png?width=231&format=png&auto=webp&s=7abcc61ae41cc10a5fed24912df0419fd3e4e849
It has been tested and verified to be correct. (For p=0.01/a=80 anyways). (Fun fact, this is the Geometric Distribution's CDF
... keep reading on reddit ➡A large sports equipment company is having a sale on both basketballs and hockey sticks. They will sell the basketballs for $20 each and the hockey sticks for $10 each. Due to restrictions to their supply, they can sell no more than 1 million basketballs, nor can they sell more than 500 thousand hockey sticks. They predict their joint probability density function for their sales of basketballs x and hockey sticks y to be
p(x,y) = C xy^2
where x and y are measured in hundred-thousands of balls and sticks respectively.
Find the value of C.
I'm reading a book on machine learning. I encountered with this expression:
>The concept of probability for discrete variables can be ex- tended to that of a probability density p(x) over a continuous variable x and is such that the probability of x lying in the inter val (x,x+δx) is given by p(x)δx for δx → 0. The probability density can be expressed as the derivative of a cumulative distribution function P(x).
A graph is given, which I couldn't put here, of P(x) and p(x). And at some point at the x interval δx is marked with a green line until p(x) graph.
What is the concept of δx here?
If the density function of a continuous probability distribution is f(x)=1/x^2 for x in (0, infinity). How can we find the mean and variance since it diverges?
