\end{align}\]. If \(F(x)>G(x)\) for some \(x\in \mathbb{R}\) and \(F(x)0.5\) if alternative = "less" (rejection for small values of (6.22)) and the two equivalent two-sided alternatives \(H_1:\mathbb{P}[X \boldsymbol{\geq}Y]\neq 0.5\) or \(H_1:\mathbb{P}[X \boldsymbol{\leq}Y]\neq 0.5\) if alternative = "two.sided" (default; rejection for large and small values of (6.22)).↩︎, Equivalently, it evaluates if the main mass of probability of \(X-Y\) is located further from \(0\).↩︎, One- and two-sided alternatives are also possible, analogously to the unpaired case.↩︎, We postpone until the end of the section the treatment of the paired sample case.↩︎, If repetitions were allowed, the distributions would be be degenerate. Statistic definition. We'll see that it can actually quantify sort of an overall shift in the values up or down in one sample compared to the other. But also, when comparing sample distributions, this can be a reasonable measure of the overall differences in these distributions as an estimate of the underlying difference in this population distributions. Distribution Needed for Hypothesis Testing. The one-sided hypothesis in which \(H_1:F\leq G\) (or \(H_1:F\geq G\)) is also very relevant. Found inside – Page 158Slakter , M . J . A comparison of Pearson chi - square and Kolmogorov goodness - of - fit tests with respect to validity . ... On the estimation of the discrepancy between empirical curves of distribution for two independent samples . That is, one can conduct the test \(H_0:F=F_0\) vs. \(H_1:F\leq F_0\) (or \(H_1:F\geq F_0\)) using alternative = "less" (or alternative = "greater").↩︎, Observe that \(F=G\) is not employed to weight the integrand (6.16), as was the case in (6.4) with \(\mathrm{d}F_0(x)\). How should we compare continuous variables of multiple groups: when non-normally distributed group is present? D_{n,m}^-=&\,\max(D_{n,m,1}^-,D_{n,m,2}^-),\\ And we also have the difference means between physicians in the Midwest and the physicians in the West. Group 1 – group 2 is plotted along the y-axis for each decile (white disks), as … If you take ordinal variables and perform an operation with a deterministic outcome, you can use simple linear regression, or a sommers d test. Probability distributions of continuous and discrete variables. \(X\) is locally stochastically greater than \(Y\) in \((-\infty,-0.75)\). Two distributions that differ in spread A Kernel density estimates for the groups. \end{align}\]. Found inside – Page 142To plot lognormal distributions, in R-Commander click Distributions → Continuous distributions → Lognormal distributions ... In general, Q–Q plots compare two distributions by plotting the quantiles of one distribution against the ... It looks like you have a clear understanding of all the available tests. What I would suggest is if you would get the book, "Goodness-of-Fit-Techn... Given \((X_1,Y_1),\ldots,(X_n,Y_n)\sim F\), it tests \(H_0: F_X=F_Y\) vs. \(H_1:\mathbb{P}[X\geq Y]>0.5\), where \(F_X\) and \(F_Y\) are the marginal cdfs of \(X\) and \(Y\). So if we were to compute this numerically and relegate it to a single number, we could take the difference in average weights for males compared to females. So this reference cancels and we're left with the difference between the Southern regions and the Midwest region. Found inside – Page 75515.19 Suppose that Y1, Y2 ,..., Yn is a random sample from a continuous distribution function F(y). ... 15.5 Using Ranks for Comparing Two Population Distributions: Independent Random Samples A statistical test for comparing two ... Highlights and caveats. This text presents a comprehensive treatment of basic statistical methods and their applications. It focuses on the analysis of variance and regression, but also addressing basic ideas in experimental design and count data. \end{align*}\]. We use the student’s t distribution when comparing means when we do not know the standard deviation of the population and must estimate it from the sample. Found inside – Page 70UDC 621.391.82 ON THE THEORY OF AMPLITUDE DISTRIBUTION OF IMPULSIVE RANDOM NOISE AND ITS APPLICATION TO THE ... distributions are considered according to the two typical cases of discrete and continuous spatial distribution and are ... But it's not perhaps as clear as when we can see the clear at least median differences in the box plots. \end{align}\]. T_{n,m}(Z_{\sigma(1)},\ldots,Z_{\sigma(n+m)})\stackrel{d}{=}T_{n,m}(Z_1,\ldots,Z_{n+m}) D_{n,m,2}:=&\,\sqrt{\frac{nm}{n+m}}\max_{1\leq j\leq m}\left|F_n(Y_{(j)})-\frac{j}{m}\right|.\nonumber Is there a reason you're specifically using a non-parametric test instead of a classic t-test? We summarize next the whole permutation-based procedure for performing a homogeneity test that rejects \(H_0\) for large values of the test statistic: Compute \(T_{n,m}\equiv T_{n,m}(Z_1,\ldots,Z_{n+m})\). Even if in reality x1 is stochastically greater than, # y1 (so H0 is false), the alternative to which H1 is confronted is even less, # plausible. Statistic computation. It uses the KL divergence to calculate a normalized score that is symmetrical. Implementation in R. For continuous data and continuous \(F=G\), the test statistic \(D_{n,m}^-\) and the asymptotic \(p\)-value are readily available through the ks.test function. If \(B=10^4\) and \(N=18!\) (\(n=m=9\)), this probability is \(7.81\times 10^{-9}\).↩︎, This sampling is done by extracting, without replacement, elements from the pooled sample z <- c(x, y). \lim_{n,m\to\infty}\mathbb{P}[D_{n,m}\leq x]=K(x). One must specify alternative = "greater" to test \(H_0\) against \(H_1:\mathbb{P}[X \boldsymbol{\geq}Y]>0.5\).246 The exact cdf of \(S_n\) under \(H_0\) is available in psignrank. Course 1 of 4 in the Biostatistics in Public Health Specialization. Therefore, \[\begin{align} So all of the data is continuous except for one parameter, which takes integer values from 0-9. Otherwise, it is a nonparametric extension of the paired \(t\)-test that evaluates if there is a shift in the main mass of probability of \(X\) or \(Y\) when both are related.245 The test is distribution-free. \end{align}\]. The two-sample Cramér–von Mises test is also distribution-free if \(F_0\) is continuous and the sample has no ties. Certainly the direction of comparison is arbitrary, we could've just as easily compared the females to the males and taken the difference female to male. \end{align}\], Values of \(U_n\) that are significantly larger than \((nm)/2\), the expected value under \(H_0\), indicate evidence in favor of \(H_1\). Note that the variances of \(X\) and \(Y\) are common; compare this situation with Figure 6.8. Seems like a chi square test might be the way to go? Whereas the remaining 9,000 plus are greater than 40 years old. The distribution of a variable is a description of the frequency of occurrence of each possible outcome. This characterization informs on the closest the Wilcoxon–Mann–Whitney test gets to a nonparametric version of the \(t\)-test.242 However, as the two counterexamples in Figures 6.9 and 6.10 respectively illustrate, (6.21) is not true in general, neither for means nor for medians. Here’s the problem: there are two kinds of variables — continuous and categorical (sometimes called discrete or factor variables) and hence, we need a … For context, I have two populations of “flows” (from spacecraft data), each with a number of measured parameters. The null distribution of \(T_{n,m}\) can be approximated through (6.28). Computing (6.25) is straightforward. All in all, 10,000 random samples were generated for each of the 98 distribution combinations! If \(H_0:F=G\) holds, then \(D_{n,m}\) tends to be small. A first pass at a plot might involve using a histogram with fill to represent the different parties. 10 3 Using statistics to compare two distributions - YouTube A_{n,m}^2:=\frac{nm}{n+m}\int\frac{(F_n(z)-G_m(z))^2}{H_{n+m}(z)(1-H_{n+m}(z))}\,\mathrm{d}H_{n+m}(z),\tag{6.18} Found inside – Page 65Therefore the purpose of this study was to compare two continuous granulators: a statistical design of experiments ... results for particle size distribution and friability of dicalcium phosphate and lactose granules are shown in Tab. If there are some combinations of location and approach that are rare, you might want the Fisher exact test instead. Statistic definition. Conversely, when \(F\neq G\), larger values of \(D_{n,m}\) are expected, and the test rejects \(H_0\) when \(D_{n,m}\) is large. Therefore, although hard to detect, the two-sample Kolmogorov–Smirnov test should eventually reject \(H_0:F=G\) in favor of \(H_1:F\leq G\). If , are continuous random variables (defined on the same probability space) then their joint pdf is a function such that. If \(H_0\) holds and \(F=G\) is continuous, then \(A_{n,m}^2\) has the same asymptotic cdf as \(A_n^2\) (see (6.8)). That said, you can also use the t test to test for differences in ordinal data, such as 0 is "worst" and 9 is "best", and this turns out to be a powerful and interpretable result. And I would argue that now with the skewed data it's difficult to see visually what's going on here when we compare stacked histograms. Many -statistical test are based upon the assumption that the data are sampled from a Gaussian distribution. Implementation in R. See below for the statistic implementation. Precisely, the probability is pbirthday(n = B, classes = factorial(N)). Given \(X_1,\ldots,X_n\sim F\) and \(Y_1,\ldots,Y_m\sim G\), it tests \(H_0: F=G\) vs. \(H_1: F\neq G\) consistently against all the alternatives in \(H_1\). We can consider the pdf for two random variables (or more). Enter the “permutation world.” For \(b=1,\ldots,B\): \[\begin{align*} Probabilities of continuous random variables (X) are defined as the area under the curve of its PDF. That's why you're testing them. The average difference after leveling the playing field in terms of these other things between men and women it was $13,400, US dollars per year which is very sizable. Let's look at our length of stay data, but we're going to split it out by age of first claim in the year 2011 for all subjects in the Heritage Health Plan who had in-patient stays of at least one day in the year 2011. And here they present the mean salary for physicians from each of these four regions. Assume that two iid samples \(X_1,\ldots,X_n\) and \(Y_1,\ldots,Y_m\) arbitrary distributions \(F\) and \(G\) are given. A_{n,m}^2=\frac{nm}{(n+m)^2}\sum_{k=1}^{n+m-1}\frac{(F_n(Z_{(k)})-G_m(Z_{(k)}))^2}{H_{n+m}(Z_{(k)})(1-H_{n+m}(Z_{(k)}))}. For more than two categories, you might want to omit the histograms and just overlay the density estimates. Continuous means that the variable can take on any reasonable value. Testing for normality. To find this we can say, on average male children weigh more than female children by 0.7 kilograms. Hence, \(H_1:\mathbb{P}[X\geq Y]>0.5\) is more specific than “\(X\) is stochastically greater than \(Y\).” \(H_1\) neither implies nor is implied by the two-sample Kolmogorov–Smirnov’s alternative \(H_1':F\leq G\) (which can be regarded as local stochastic dominance).241. The variable that you care about (and want to see if it is different between the two groups) must be continuous. The sampling distribution of proportion means. To get a difference in average salaries between physicians in the South and physicians in the Midwest of negative $4,451. And I'm sure you can come up with questions of interest that you have. You will learn how to create tables and histograms of each type of data. \end{align*}\], The converse implication is false (see Figure 6.8). Although if you look carefully you can see that the tail perhaps shifts over a bit for those who are older than 40 relative to those who are less than 40. The differences in the two data points for each of the categories range from 0 to 20. Given \(X_1,\ldots,X_n\sim F\) and \(Y_1,\ldots,Y_m\sim G\), it tests \(H_0: F=G\) vs. \(H_1: F\leq G\).230 Rejection of \(H_0\) in favor of \(H_1\) gives evidence for the local stochastic dominance of \(F\) by \(G\) (which may or may not be global). What we actually have about these respective regions? The pairs of samples are analyzed using both the two sample t-test and the Mann-Whitney test to compare how well each test performs. Found inside – Page 423Table of the hyperbolic transformation sinh -'Vx , Journal of the Royal Statistical Society , Series A , 113 , 228-229 . Aspin , A. ( 1948 ) . An examination and further development of a formula arising in the problem of comparing two ... Highlights and caveats. Found inside – Page 62Another way to compare two groups on a continuous measure is the non-parametric Wilcoxon-Mann-Whitney Test (WMW). ... If the distributions of the two groups differ only in location (i.e. they have the same shape), then this is a test ... Test purpose. Permutations (see Section 6.2.3) can be used for obtaining non-asymptotic \(p\)-values and dealing with discrete samples. \(X\) is not stochastically greater than \(Y\), as the cdfs cross, but \(\mathbb{P}\lbrack X\geq Y\rbrack=0.834\). A_{n,m}^2=\frac{nm}{(n+m)^2}\sum_{k=1}^{n+m}\frac{(F_n(Z_k)-G_m(Z_k))^2}{H_{n+m}(Z_k)(1-H_{n+m}(Z_k))}.\tag{6.19} &\iff \mathbb{P}[X>x]\geq \mathbb{P}[Y>x].\tag{6.14} \mathbb{P}[X\geq Y]>0.5 \iff m_X>m_Y.\tag{6.21} Found inside – Page 20Figure 2.2 : Use of a random number j chosen from a uniform distribution ( 0,1 ) to find a random number x from a distribution with cumulative distribution function F ( x ) 1 ( a ) F ( x ) Continuous distribution u 0 X A x = F - 1 ( U ) ... In general, I advice away from statistical tests, but not everyone has caught up with this notion (and it isn't practical many times---but from a lack of organizational knowledge and not the power of the technique). \end{align*}\]. W_{n,m}^2:=\frac{nm}{n+m}\int(F_n(z)-G_m(z))^2\,\mathrm{d}H_{n+m}(z),\tag{6.16} We could take the difference in medians between two sample distributions. Discrete Random Variables 3. Averages 4. Bernoulli and Related Variables 5. Continuous Random Variables 6. Families of Continuous Distributions 7. Organizing and Describing Data 8. Samples, Statistics, and Sampling Distributions 9. Ideally I’d like to apply the same test to all my parameters. Here \(\hat{S}_X^2\) and \(\hat{S}_Y^2\) represent the quasi-variances of \(X_1,\ldots,X_n\) and \(Y_1,\ldots,Y_m\), respectively. Pictured are two distributions of data, X 1 and X 2, with unknown means and standard deviations.The second panel shows the sampling distribution of the newly created random variable ().This distribution is the theoretical distribution of many many sample means from population 1 … To do so: Among the previous ecdf-based tests of homogeneity, only the two-sample Kolmogorov–Smirnov test was able to readily deal with one-sided alternatives. Our intention in preparing this book was to present in as simple a manner as possible those branches of error analysis which ?nd direct applications in solving various problems in engineering practice. In a very vague and imprecise form, these tests can be interpreted as “nonparametric \(t\)-tests” for unpaired and paired data.239 The rationale is that, very often, the aforementioned one-sided alternatives are related to differences in the distributions produced by a shift in their main masses of probability. Then it is observed that the probability density function ƒ(x) = dF(x)/dx and that ∫ ƒ(x) dx = 1. The graph combines the first two rows of the panel in the previous section. \end{align}\]. So for example, if we make West the reference region, then the three mean differences we could report are the mean differences between the Midwest and the West. And then looked at whether the sex distribution of the physicians differed by region as well. \(H_{n+m}(z)=\frac{n}{n+m}F_n(z)+\frac{m}{n+m}G_m(z)\), \(\mathbb{P}\lbrack X\geq Y\rbrack=0.834\), \(Y\sim0.8\mathcal{N}(-3,1)+0.2\mathcal{N}(5,4)\), \(\mathbb{P}\lbrack X\geq Y\rbrack=0.5356\), \(X\sim0.55\mathcal{N}(-1.5,1.25^2)+0.45\mathcal{N}(7,0.75^2)\), \(Y\sim0.4\mathcal{N}(-5,1)+0.6\mathcal{N}(5,1)\), \(\mathbb{P}\lbrack X\geq Y\rbrack=0.6520\), \(\mathrm{rank}_{X,Y}(X_i):=nF_n(X_i)+mG_n(X_i)\), \(\mathbb{V}\mathrm{ar}[U_{n;\mathrm{MW}}]=nm(n+m+1)/12\), \(H_1:\mathbb{P}[X \boldsymbol{\geq}Y]>0.5\), ## Wilcoxon rank sum test with continuity correction, ## alternative hypothesis: true location shift is not equal to 0, ## alternative hypothesis: true location shift is greater than 0, ## alternative hypothesis: true location shift is less than 0, # Beware! The reason is because \(F=G\) is unknown. Highlights and caveats. Because there's more detail in each histogram when it comes to comparing the distributions, it's a little harder to see and make generalizations. If \(F\geq G\), then \(X\) is stochastically smaller than \(Y\) (equivalently, \(Y\) is stochastically greater than \(X\)).↩︎, With this terminology, clearly global stochastic dominance implies local stochastic dominance, but not the other way around.↩︎, \(G_m\) is the ecdf of \(Y_1,\ldots,Y_m\).↩︎, Asymptotic when the sample sizes \(n\) and \(m\) are large: \(n,m\to\infty\).↩︎, The case \(H_1:F\geq G\) is analogous and not required, as the roles of \(F\) and \(G\) can be interchanged.↩︎, Analogously, \(D_{n,m,j}^+\) changes the sign inside the maximum of \(D_{n,m,j}^-\), \(j=1,2\), and \(D_{n,m}^+:=\max(D_{n,m,1}^+,D_{n,m,2}^+)\).↩︎, Confusingly, ks.test’s alternative does not refer to the direction of stochastic dominance, which is the codification used in the alternative arguments of the t.test and wilcox.test functions.↩︎, Observe that ks.test also implements the one-sample Kolmogorov–Smirnov test seen in Section 6.1.1 with one-sided alternatives. You could use a Cramer-von Mises statistic (or anything similar to it that you think makes sense) with a discrete distribution as long as you treat... \end{align}\], where \(\hat{\sigma}_b\), \(b=1,\ldots,B\), denote the \(B\) randomly-chosen \((n+m)\)-permutations.250. Some key approaches include visual comparisons, such as these side by side box plots, and numerical comparisons, mainly the mean difference between any two groups of samples. For discrete \(F=G\), a test implementation can be achieved through the permutation approach to be seen in Section 6.2.3. Designating the region is arbitrary, just like the direction of comparison when we have two groups is as well because essentially we're designating the reference there as well. However, the evaluation of the sum of \((n+m)!\) terms is often impossible. Exercise 6.11 Verify the correctness of the asymptotic null distributions of \(W_{n,m}^2\) and \(A_{n,m}^2\). \mathbb{P}[T_{n,m}\leq x]\approx \frac{1}{B}\sum_{b=1}^B 1_{\big\{T^{\hat{\sigma}_b}_{n,m}\leq x\big\}}, \tag{6.28} Requires TWO arguments, one being the original, # data (X_1, ..., X_n, Y_1, ..., Y_m) and the other containing the random, # Perform permutation resampling with the aid of boot::boot, "Permutation-based Anderson-Darling test of homogeneity", # p-value: modify if rejection does not happen for large values of the, # Plot the position of the original statistic with respect to the, ## Permutation-based Anderson-Darling test of homogeneity, ## alternative hypothesis: any alternative to homogeneity, \((X_1^{*b},Y_1^{*b}),\ldots,(X_n^{*b},Y_n^{*b})\), “On a Test of Whether One of Two Random Variables Is Stochastically Larger Than the Other.”, “Individual Comparisons by Ranking Methods.”, \(D_{n,m}^+:=\max(D_{n,m,1}^+,D_{n,m,2}^+)\), \((\bar{X} -\bar{Y})/\sqrt{(\hat{S}_X^2/n+\hat{S}_Y^2/m)} \stackrel{d}{\longrightarrow}\mathcal{N}(0,1)\), \(\bar{X}/(\hat{S}/\sqrt{n}) \stackrel{d}{\longrightarrow}\mathcal{N}(0,1)\), \(H_1:\mathbb{P}[X \boldsymbol{\leq}Y]>0.5\), \(H_1:\mathbb{P}[X \boldsymbol{\geq}Y]\neq 0.5\), \(H_1:\mathbb{P}[X \boldsymbol{\leq}Y]\neq 0.5\), estimated number of atoms in the observable universe, Once you have a working solution, increase. \(X\sim F\) is not stochastically greater than \(Y\sim G\). In summary: I'm trying to compare the distribution of two samples containing measurements with continuous likelihood distributions rather than precise measurements. This is an article we'll look at throughout the course because it gives some useful examples of techniques we'll be doing in the course in a nice context. Found inside – Page 32The hypergeometric distribution is the fundamental basis to construct the log-rank test when comparing two survival curves. Besides the discrete random variables, many continuous random variables also play an important role in clinical ... The direction of stochastic dominance is opposite to the direction of dominance of the cdfs. A continuous distribution describes the probabilities of the possible values of a continuous random variable. This means that the divergence of P from Q is the same as Q from P, or stated formally: The ecdf-based goodness-of-fit tests seen in Section 6.1.1 can be adapted to the homogeneity problem, with varying difficulty and versatility. Perform tests of a population mean using a normal distribution or a Student’s t-distribution. They adjusted for a bunch of other things that included academic rank, leadership position, publication numbers, etc. A p-value ~ 1 indicates one is probably conducting the test in. If \(H_0\) holds and \(F=G\) is continuous, then \(W_{n,m}^2\) has the same asymptotic cdf as \(W_n^2\) (see (6.6)). And so whittling down the comparison to a single number summary will allow us to create a confidence interval on this summary measure. In our earlier example with age and income distributions, we compared a sample distribution to another sample distribution instead of a theoretical distribution. In this case, we need to apply resampling techniques such as permutation tests or bootstrapping to derive a KS test statistic distribution. You've specified that observer A sees a normal distribution, in other words you're saying. where we use the standard notation for the pooled sample, \(Z_i=X_i\) and \(Z_{j+n}=Y_j\) for \(i=1,\ldots,n\) and \(j=1,\ldots,m\). Which is essentially the same as stating, like we did before, on average male children weigh more than female children by 0.7 kilograms. Biostatistics is the application of statistical reasoning to the life sciences, and it is the key to unlocking the data gathered by researchers and the evidence presented in the scientific literature. Statistic definition. dwilcox considers (m, n) as the sample sizes of (X, Y), \(\mu_X>\mu_Y \implies \mathbb{P}[X\geq Y]>0.5\), \(m_X>m_Y \implies \mathbb{P}[X\geq Y]>0.5\), ## Wilcoxon signed rank test with continuity correction, \(X\sim 0.5\mathcal{N}(-\delta,(1 + 0.25 \delta)^2)+0.5\mathcal{N}(\delta,(1 + 0.25 \delta)^2)\), \(\sigma:\{1,\ldots,N\}\longrightarrow \{1,\ldots,N\}\), \(T_{n,m}\equiv T_{n,m}(Z_1,\ldots,Z_{n+m})\), \(T^{\sigma}_{n,m}\equiv T_{n,m}(Z_{\sigma(1)},\ldots,Z_{\sigma(n+m)})\), \(\mathbb{P}[T_{n,m}\leq x] = \mathbb{P}[T^\sigma_{n,m}\leq x]\), \(1_{\big\{T^{\sigma_k}_{n,m}\leq x\big\}}\), \(T_{n,m}^{*b}\equiv T_{n,m}(Z_1^{*b},\ldots,Z_{n+m}^{*b})\), # A homogeneity test using the Anderson-Darling statistic, # Test statistic function. You could still use a t-test, depending on the distribution. This volume consists of three parts: Part I comprises 11 chapters on the basic concepts of statistics, Part II consists of 10 chapters on multivariate statistics and Part III contains 12 chapters on design and analysis for medical research. Experiment with your discrete parameter, and see how to best include it in the model. If not, then why? And we can see that if we look at the distribution of weights in kilograms for male and female children, males tend to weigh more than females. The formula (6.19) is reasonably direct for computation. You can have a look on optimal transport between distributions. It is often used in Computer Vision (under the name Earth Mover Distance) to comp... Details. So in summary, while the distributions of continuous data can be compared between samples in many ways. The comparison of \(F\) and \(G\) can be done by testing their equality, which is known as the testing for homogeneity of the samples \(X_1,\ldots,X_n\) and \(Y_1,\ldots,Y_m\). Comparing statistical distributions Students are expected to be able to compare data sets by considering graphs, averages and measures of spread. I’m making this blog post mainly because many of the options I will show can’t be done in SPSS directly … It would be negative 0.7 when we compare females to males because females weigh less on average. Statistic definition. So you have the chi-square test to compare the discrete distributions (if you want to be nonparametric). Conversely, when \(H_0:F=G\) or \(F\geq G\) holds, smaller values of \(D_{n,m}^-\) are expected, possibly negative. Statistics, Normal Distribution, summary measures, binary data. The group of \(41\) students contains two subgroups in mf. And I'll do it in the direction of taking the mean for those who are greater than 40 years minus the mean for those than less than 40 years. velocity), and so I can use something like the Kolmogorov-Smirnov test to compare the two populations. The probability distributions available in the Stochastic Element are mainly theoretical continuous distributions which are defined by entering or providing the two or three parameters necessary to completely describe the distribution.
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