As usual, we will let \(G\) denote the distribution function of \(Y\) and \(g\) the probability density function of \(Y\). \(X\) is uniformly distributed on the interval \([-1, 3]\). More generally, it's easy to see that every positive power of a distribution function is a distribution function. \(f(u) = \left(1 - \frac{u-1}{6}\right)^n - \left(1 - \frac{u}{6}\right)^n, \quad u \in \{1, 2, 3, 4, 5, 6\}\), \(g(v) = \left(\frac{v}{6}\right)^n - \left(\frac{v - 1}{6}\right)^n, \quad v \in \{1, 2, 3, 4, 5, 6\}\). Thus suppose that \(\bs X\) is a random variable taking values in \(S \subseteq \R^n\) and that \(\bs X\) has a continuous distribution on \(S\) with probability density function \(f\). It is possible that your data does not look Gaussian or fails a normality test, but can be transformed to make it fit a Gaussian distribution. For the following three exercises, recall that the standard uniform distribution is the uniform distribution on the interval \( [0, 1] \). \(g(t) = a e^{-a t}\) for \(0 \le t \lt \infty\) where \(a = r_1 + r_2 + \cdots + r_n\), \(H(t) = \left(1 - e^{-r_1 t}\right) \left(1 - e^{-r_2 t}\right) \cdots \left(1 - e^{-r_n t}\right)\) for \(0 \le t \lt \infty\), \(h(t) = n r e^{-r t} \left(1 - e^{-r t}\right)^{n-1}\) for \(0 \le t \lt \infty\). Recall that if \((X_1, X_2, X_3)\) is a sequence of independent random variables, each with the standard uniform distribution, then \(f\), \(f^{*2}\), and \(f^{*3}\) are the probability density functions of \(X_1\), \(X_1 + X_2\), and \(X_1 + X_2 + X_3\), respectively. That is, \( f * \delta = \delta * f = f \). Find the probability density function of each of the following random variables: In the previous exercise, \(V\) also has a Pareto distribution but with parameter \(\frac{a}{2}\); \(Y\) has the beta distribution with parameters \(a\) and \(b = 1\); and \(Z\) has the exponential distribution with rate parameter \(a\). Moreover, this type of transformation leads to simple applications of the change of variable theorems. Case when a, b are negativeProof that if X is a normally distributed random variable with mean mu and variance sigma squared, a linear transformation of X (a. Suppose again that \((T_1, T_2, \ldots, T_n)\) is a sequence of independent random variables, and that \(T_i\) has the exponential distribution with rate parameter \(r_i \gt 0\) for each \(i \in \{1, 2, \ldots, n\}\). On the other hand, \(W\) has a Pareto distribution, named for Vilfredo Pareto. We have seen this derivation before. \(g(u, v, w) = \frac{1}{2}\) for \((u, v, w)\) in the rectangular region \(T \subset \R^3\) with vertices \(\{(0,0,0), (1,0,1), (1,1,0), (0,1,1), (2,1,1), (1,1,2), (1,2,1), (2,2,2)\}\). For the next exercise, recall that the floor and ceiling functions on \(\R\) are defined by \[ \lfloor x \rfloor = \max\{n \in \Z: n \le x\}, \; \lceil x \rceil = \min\{n \in \Z: n \ge x\}, \quad x \in \R\]. Given our previous result, the one for cylindrical coordinates should come as no surprise. The linear transformation of a normally distributed random variable is still a normally distributed random variable: . Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Recall that the standard normal distribution has probability density function \(\phi\) given by \[ \phi(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^2}, \quad z \in \R\]. \sum_{x=0}^z \frac{z!}{x! These can be combined succinctly with the formula \( f(x) = p^x (1 - p)^{1 - x} \) for \( x \in \{0, 1\} \). Initialy, I was thinking of applying "exponential twisting" change of measure to y (which in this case amounts to changing the mean from $\mathbf{0}$ to $\mathbf{c}$) but this requires taking . Random variable \( V = X Y \) has probability density function \[ v \mapsto \int_{-\infty}^\infty f(x, v / x) \frac{1}{|x|} dx \], Random variable \( W = Y / X \) has probability density function \[ w \mapsto \int_{-\infty}^\infty f(x, w x) |x| dx \], We have the transformation \( u = x \), \( v = x y\) and so the inverse transformation is \( x = u \), \( y = v / u\). This chapter describes how to transform data to normal distribution in R. Parametric methods, such as t-test and ANOVA tests, assume that the dependent (outcome) variable is approximately normally distributed for every groups to be compared. Another thought of mine is to calculate the following. Recall that a Bernoulli trials sequence is a sequence \((X_1, X_2, \ldots)\) of independent, identically distributed indicator random variables. Keep the default parameter values and run the experiment in single step mode a few times. Using your calculator, simulate 5 values from the Pareto distribution with shape parameter \(a = 2\). the linear transformation matrix A = 1 2 While not as important as sums, products and quotients of real-valued random variables also occur frequently. The distribution is the same as for two standard, fair dice in (a). \(U = \min\{X_1, X_2, \ldots, X_n\}\) has distribution function \(G\) given by \(G(x) = 1 - \left[1 - F(x)\right]^n\) for \(x \in \R\). A multivariate normal distribution is a vector in multiple normally distributed variables, such that any linear combination of the variables is also normally distributed. If we have a bunch of independent alarm clocks, with exponentially distributed alarm times, then the probability that clock \(i\) is the first one to sound is \(r_i \big/ \sum_{j = 1}^n r_j\). Linear transformation of multivariate normal random variable is still multivariate normal. (iv). Related. Once again, it's best to give the inverse transformation: \( x = r \sin \phi \cos \theta \), \( y = r \sin \phi \sin \theta \), \( z = r \cos \phi \). Suppose that \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the standard uniform distribution. = f_{a+b}(z) \end{align}. Then \(\bs Y\) is uniformly distributed on \(T = \{\bs a + \bs B \bs x: \bs x \in S\}\). Let \(\bs Y = \bs a + \bs B \bs X\), where \(\bs a \in \R^n\) and \(\bs B\) is an invertible \(n \times n\) matrix. Recall that the sign function on \( \R \) (not to be confused, of course, with the sine function) is defined as follows: \[ \sgn(x) = \begin{cases} -1, & x \lt 0 \\ 0, & x = 0 \\ 1, & x \gt 0 \end{cases} \], Suppose again that \( X \) has a continuous distribution on \( \R \) with distribution function \( F \) and probability density function \( f \), and suppose in addition that the distribution of \( X \) is symmetric about 0. This is one of the older transformation technique which is very similar to Box-cox transformation but does not require the values to be strictly positive. \(\left|X\right|\) has distribution function \(G\) given by \(G(y) = F(y) - F(-y)\) for \(y \in [0, \infty)\). We introduce the auxiliary variable \( U = X \) so that we have bivariate transformations and can use our change of variables formula. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The associative property of convolution follows from the associate property of addition: \( (X + Y) + Z = X + (Y + Z) \). In this case, \( D_z = [0, z] \) for \( z \in [0, \infty) \). The Pareto distribution is studied in more detail in the chapter on Special Distributions. But a linear combination of independent (one dimensional) normal variables is another normal, so aTU is a normal variable. Systematic component - \(x\) is the explanatory variable (can be continuous or discrete) and is linear in the parameters. Using the change of variables theorem, If \( X \) and \( Y \) have discrete distributions then \( Z = X + Y \) has a discrete distribution with probability density function \( g * h \) given by \[ (g * h)(z) = \sum_{x \in D_z} g(x) h(z - x), \quad z \in T \], If \( X \) and \( Y \) have continuous distributions then \( Z = X + Y \) has a continuous distribution with probability density function \( g * h \) given by \[ (g * h)(z) = \int_{D_z} g(x) h(z - x) \, dx, \quad z \in T \], In the discrete case, suppose \( X \) and \( Y \) take values in \( \N \). This is a very basic and important question, and in a superficial sense, the solution is easy. Next, for \( (x, y, z) \in \R^3 \), let \( (r, \theta, z) \) denote the standard cylindrical coordinates, so that \( (r, \theta) \) are the standard polar coordinates of \( (x, y) \) as above, and coordinate \( z \) is left unchanged. From part (a), note that the product of \(n\) distribution functions is another distribution function. Then \[ \P(Z \in A) = \P(X + Y \in A) = \int_C f(u, v) \, d(u, v) \] Now use the change of variables \( x = u, \; z = u + v \). In particular, it follows that a positive integer power of a distribution function is a distribution function. Suppose that \(X\) has the probability density function \(f\) given by \(f(x) = 3 x^2\) for \(0 \le x \le 1\). Vary the parameter \(n\) from 1 to 3 and note the shape of the probability density function. Suppose that \((X_1, X_2, \ldots, X_n)\) is a sequence of indendent real-valued random variables and that \(X_i\) has distribution function \(F_i\) for \(i \in \{1, 2, \ldots, n\}\). As with the above example, this can be extended to multiple variables of non-linear transformations. Returning to the case of general \(n\), note that \(T_i \lt T_j\) for all \(j \ne i\) if and only if \(T_i \lt \min\left\{T_j: j \ne i\right\}\). Formal proof of this result can be undertaken quite easily using characteristic functions. \(\bs Y\) has probability density function \(g\) given by \[ g(\bs y) = \frac{1}{\left| \det(\bs B)\right|} f\left[ B^{-1}(\bs y - \bs a) \right], \quad \bs y \in T \]. These results follow immediately from the previous theorem, since \( f(x, y) = g(x) h(y) \) for \( (x, y) \in \R^2 \). Set \(k = 1\) (this gives the minimum \(U\)). This page titled 3.7: Transformations of Random Variables is shared under a CC BY 2.0 license and was authored, remixed, and/or curated by Kyle Siegrist (Random Services) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. This follows from the previous theorem, since \( F(-y) = 1 - F(y) \) for \( y \gt 0 \) by symmetry. When plotted on a graph, the data follows a bell shape, with most values clustering around a central region and tapering off as they go further away from the center. Linear transformations (addition and multiplication of a constant) and their impacts on center (mean) and spread (standard deviation) of a distribution. More generally, if \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the standard uniform distribution, then the distribution of \(\sum_{i=1}^n X_i\) (which has probability density function \(f^{*n}\)) is known as the Irwin-Hall distribution with parameter \(n\). Using your calculator, simulate 5 values from the exponential distribution with parameter \(r = 3\). How could we construct a non-integer power of a distribution function in a probabilistic way? Suppose that \( (X, Y) \) has a continuous distribution on \( \R^2 \) with probability density function \( f \). Also, a constant is independent of every other random variable. Find the probability density function of \(X = \ln T\). For \( z \in T \), let \( D_z = \{x \in R: z - x \in S\} \). Open the Cauchy experiment, which is a simulation of the light problem in the previous exercise. Using your calculator, simulate 5 values from the uniform distribution on the interval \([2, 10]\). Suppose first that \(X\) is a random variable taking values in an interval \(S \subseteq \R\) and that \(X\) has a continuous distribution on \(S\) with probability density function \(f\). As usual, we start with a random experiment modeled by a probability space \((\Omega, \mathscr F, \P)\). Recall that the Poisson distribution with parameter \(t \in (0, \infty)\) has probability density function \(f\) given by \[ f_t(n) = e^{-t} \frac{t^n}{n! This subsection contains computational exercises, many of which involve special parametric families of distributions. The Jacobian is the infinitesimal scale factor that describes how \(n\)-dimensional volume changes under the transformation. \(\P(Y \in B) = \P\left[X \in r^{-1}(B)\right]\) for \(B \subseteq T\). and a complete solution is presented for an arbitrary probability distribution with finite fourth-order moments. The grades are generally low, so the teacher decides to curve the grades using the transformation \( Z = 10 \sqrt{Y} = 100 \sqrt{X}\). MULTIVARIATE NORMAL DISTRIBUTION (Part I) 1 Lecture 3 Review: Random vectors: vectors of random variables. A = [T(e1) T(e2) T(en)]. This is known as the change of variables formula. In particular, the times between arrivals in the Poisson model of random points in time have independent, identically distributed exponential distributions. \(Y\) has probability density function \( g \) given by \[ g(y) = \frac{1}{\left|b\right|} f\left(\frac{y - a}{b}\right), \quad y \in T \]. The images below give a graphical interpretation of the formula in the two cases where \(r\) is increasing and where \(r\) is decreasing. The normal distribution is studied in detail in the chapter on Special Distributions. In statistical terms, \( \bs X \) corresponds to sampling from the common distribution.By convention, \( Y_0 = 0 \), so naturally we take \( f^{*0} = \delta \). As before, determining this set \( D_z \) is often the most challenging step in finding the probability density function of \(Z\). Assuming that we can compute \(F^{-1}\), the previous exercise shows how we can simulate a distribution with distribution function \(F\). However, when dealing with the assumptions of linear regression, you can consider transformations of . Let \( z \in \N \). Note that the joint PDF of \( (X, Y) \) is \[ f(x, y) = \phi(x) \phi(y) = \frac{1}{2 \pi} e^{-\frac{1}{2}\left(x^2 + y^2\right)}, \quad (x, y) \in \R^2 \] From the result above polar coordinates, the PDF of \( (R, \Theta) \) is \[ g(r, \theta) = f(r \cos \theta , r \sin \theta) r = \frac{1}{2 \pi} r e^{-\frac{1}{2} r^2}, \quad (r, \theta) \in [0, \infty) \times [0, 2 \pi) \] From the factorization theorem for joint PDFs, it follows that \( R \) has probability density function \( h(r) = r e^{-\frac{1}{2} r^2} \) for \( 0 \le r \lt \infty \), \( \Theta \) is uniformly distributed on \( [0, 2 \pi) \), and that \( R \) and \( \Theta \) are independent. cov(X,Y) is a matrix with i,j entry cov(Xi,Yj) . Then: X + N ( + , 2 2) Proof Let Z = X + . I have to apply a non-linear transformation over the variable x, let's call k the new transformed variable, defined as: k = x ^ -2. Suppose that \(X\) and \(Y\) are random variables on a probability space, taking values in \( R \subseteq \R\) and \( S \subseteq \R \), respectively, so that \( (X, Y) \) takes values in a subset of \( R \times S \). The PDF of \( \Theta \) is \( f(\theta) = \frac{1}{\pi} \) for \( -\frac{\pi}{2} \le \theta \le \frac{\pi}{2} \). Hence by independence, \[H(x) = \P(V \le x) = \P(X_1 \le x) \P(X_2 \le x) \cdots \P(X_n \le x) = F_1(x) F_2(x) \cdots F_n(x), \quad x \in \R\], Note that since \( U \) as the minimum of the variables, \(\{U \gt x\} = \{X_1 \gt x, X_2 \gt x, \ldots, X_n \gt x\}\). The computations are straightforward using the product rule for derivatives, but the results are a bit of a mess. Let A be the m n matrix Then \(U\) is the lifetime of the series system which operates if and only if each component is operating. Proof: The moment-generating function of a random vector x x is M x(t) = E(exp[tTx]) (3) (3) M x ( t) = E ( exp [ t T x]) Linear Transformation of Gaussian Random Variable Theorem Let , and be real numbers . Vary \(n\) with the scroll bar and set \(k = n\) each time (this gives the maximum \(V\)). The critical property satisfied by the quantile function (regardless of the type of distribution) is \( F^{-1}(p) \le x \) if and only if \( p \le F(x) \) for \( p \in (0, 1) \) and \( x \in \R \). When \(n = 2\), the result was shown in the section on joint distributions. The sample mean can be written as and the sample variance can be written as If we use the above proposition (independence between a linear transformation and a quadratic form), verifying the independence of and boils down to verifying that which can be easily checked by directly performing the multiplication of and . Often, such properties are what make the parametric families special in the first place. If \( X \) takes values in \( S \subseteq \R \) and \( Y \) takes values in \( T \subseteq \R \), then for a given \( v \in \R \), the integral in (a) is over \( \{x \in S: v / x \in T\} \), and for a given \( w \in \R \), the integral in (b) is over \( \{x \in S: w x \in T\} \). It must be understood that \(x\) on the right should be written in terms of \(y\) via the inverse function. Let M Z be the moment generating function of Z . Proposition Let be a multivariate normal random vector with mean and covariance matrix . The distribution of \( Y_n \) is the binomial distribution with parameters \(n\) and \(p\). In this section, we consider the bivariate normal distribution first, because explicit results can be given and because graphical interpretations are possible. In this case, \( D_z = \{0, 1, \ldots, z\} \) for \( z \in \N \). But first recall that for \( B \subseteq T \), \(r^{-1}(B) = \{x \in S: r(x) \in B\}\) is the inverse image of \(B\) under \(r\). Suppose that \( X \) and \( Y \) are independent random variables with continuous distributions on \( \R \) having probability density functions \( g \) and \( h \), respectively. The transformation is \( x = \tan \theta \) so the inverse transformation is \( \theta = \arctan x \). \( f(x) \to 0 \) as \( x \to \infty \) and as \( x \to -\infty \). Graph \( f \), \( f^{*2} \), and \( f^{*3} \)on the same set of axes. The Erlang distribution is studied in more detail in the chapter on the Poisson Process, and in greater generality, the gamma distribution is studied in the chapter on Special Distributions. Probability, Mathematical Statistics, and Stochastic Processes (Siegrist), { "3.01:_Discrete_Distributions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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