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### Intuition of pdf of a continuous random variable [duplicate]

What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if ...
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### Solving for a pdf of a function of a continuous random variable. Justification and reason for the procedure [duplicate]

Short version: When solving for the pdf of a function of a continuous random variable(say, $Y=X^2$), why can't you just plug in inverse of that($\pm\sqrt{x}$) into the pdf of the RV? Why do you have ...
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Suppose we have a discrete r.v. $X$, take $Y = g(X)$ where $g$ is one-to-one and onto- If we want to obtain the new pdf for the discrete r.v. we simply notice that $$f_Y(y) = P(Y=y) = f_X(g^{-1}(y)) ... 0answers 59 views ### Probability density function for a logarithmic spinner [duplicate] I'm having trouble deriving the PDF for the following problem (from the book Doing Bayesian analysis with R and BUGS): Consider a spinner of the kind often found with board games, but with a ... 0answers 25 views ### Where can I find a citable reference for the formula for inverse distributions? [duplicate] To be more specific: Let the random variable X be distributed according to probability density function f(x). Then the pdf of Y=1/X is given by g(y)=1/y^2 f(1/y), see wikipedia. I wonder if there is ... 1answer 6k views ### Derivation of change of variables of a probability density function? In the book pattern recognition and machine learning (formula 1.27), it gives$$p_y(y)=p_x(x) \left | \frac{d x}{d y} \right |=p_x(g(y)) | g'(y) |$$where x=g(y), p_x(x) is the pdf that ... 2answers 4k views ### Scale parameters — How do they work, why are they sometimes dropped? I'm having difficulty wrapping my head around scale parameters. How exactly do they work? Why are they sometimes ignored? (in other words, when is it important to preserve them in a calculation?) ... 2answers 3k views ### Distribution of function of variable having a Gaussian distribution If I have a variable X whose Gaussian distribution is known and let f be a known function, is there a way to compute the distribution of f(X) i.e. the resulting Gaussian distribution from this? ... 1answer 5k views ### Cosine of a uniform random variable I have to find pdf of Y = \cos(X) where X is a random variable distributed uniformly in [-\pi,\pi]. I solved this using distribution function method, and the result was: f_{Y}(y) = \dfrac{1}{\... 1answer 9k views ### how to read y axis in kernel density graph [duplicate] I need to understand how to read kernel density graphs. How do you come up with the values in y-axis? 1answer 3k views ### Different probability density transformations due to Jacobian factor In Bishop's Pattern Recognition and Machine Learning I read the following, just after the probability density p(x\in(a,b))=\int_a^bp(x)\textrm{d}x was introduced: Under a nonlinear change of ... 2answers 3k views ### Operations on probability distributions of continuous random variables How do probability distributions of continuous random variables transform under functions? I.e. I have a random variable, X, drawn from a normal distribution with mean 0 and variance 1. What is the ... 1answer 3k views ### Base-10 lognormal PDF integrated over log10(x) From what I understand, the lognormal probability density function in base-10 is mathematically defined thus:$$ p(x; \mu, \sigma) = \frac{log_{10}(e)}{x \sigma \sqrt{2 \pi}} e^{-\frac{(log_{10}(x) -...
If a random variable $W$ is Normally distributed, then $\exp(W)$ is Log-Normally distributed. However, the pdfs of these two random variables differ by a factor of $\exp(W)^{-1}$. The Normal pdf ...