Distributions in statistics pdf

Exploratory Data Analysis 1. EDA Techniques 1. Probability Distributions 1. Detailed information on a few of the most common distributions is available below. There are a large number of distributions used in statistical applications. It is beyond the scope of this Handbook to discuss more than a few of these. Two excellent sources for additional detailed information on a large array of distributions are Johnson, Kotz, and Balakrishnan and Evans, Hastings, and Peacock.

Equations for the probability functions are given for the standard form of the distribution. Formulas exist for defining the functions with location and scale parameters in terms of the standard form of the distribution. The sections on parameter estimation are restricted to the method of moments and maximum likelihood.

This is because the least squares and PPCC and probability plot estimation procedures are generic. The maximum likelihood equations are not listed if they involve solving simultaneous equations. Note that the CDF 2. Table 1. Theorem 2. Based on the generalized binomial expansion, we can write 2.

QY FR x. For several important distributions such as the Weibull, log-logistic, exponentiated-exponential and exponential listed in Table 1 and the normal, Student-t, gamma and beta distributions, among others, QY u can be expanded as in Equation 2. By application of an equation in Section 0.

Combining equations 2. Thus, some mathematical properties of this class such as the ordinary and incomplete moments and generating function can be determined by knowing those of the exp-FR distribution, which have been investigated by some authors for several baseline distributions. The following Lemma gives the relationships between the random variables R and Y for some cases, which can be used to simulate the random variable X from the random variable T.

Lemma 3. Remark 3. The following results are obtained from equations 2. Theorem 3. From 4. Next, we provide some properties of a special case of 4. Remark 4. Lemma 4. Using 4. The skewness and kurtosis of X can be determined from the ordinary moments using well-known relationships. The result follows from Lemma 4. Estimation and Simulations Several approaches for parameter estimation were proposed in the literature but the maximum like- lihood method is the most commonly employed.

The maximum likelihood estimators MLEs enjoy desirable properties and can be used when constructing confidence intervals for the model parame- ters and also in test statistics. The normal approximation for these estimators in large sample dis- tribution theory is easily handled either analytically or numerically. Let x1 ,. We consider the estimation of the unknown parameters by the maximum likelihood method. The MLEs are determined by maximizing the log-likelihood function in Equation 5.

From the results of the simulations, we can verify that the biases and MSEs decrease in general when the sample size n increases. Therefore, the MLEs and their asymptotic results can be used for estimating and constructing confidence intervals even for reason- ably small sample sizes.

Table 2. In the applications, the model parameters are estimated by the method of maximum likelihood. Uncensored data sets 6. Data set 1: Cancer data. The first data set represents the remission times in months of bladder cancer patients studied by Lee and Wang These data were used by Zea et al. The estimates and goodness-of-fit statistics of the exponentiated- Weibull EW Mudholkar and Srivastava, and Weibull models are also reported in Table 3.

For a visual comparison, we provide the PP-plots of the fitted models in Figure 3. Data set 2: Carbone Fibre Data. The second data set has recently been used by Cordeiro and Lemonte to illustrate the appli- cability of the beta-Birnbaum-Saunders BBS distribution.

We also provide estimates and goodness-of-fit statistics of the EW and Weibull models in Table 4. Further, the PP-plots in Figure 4 also support the results in Table 4. Data set 3: Aarset Data. The third data set is taken from Aarset which represents the lifetimes of 50 devices.

Recently, Silva et al. Furthermore, the PP-plots in Figure 5 also support the results in Table 5. Table 3. The indicator ri is equal to 1 if a failure is observed and 0 otherwise. Suppose that the data are independently and identically distributed and come from a distribution with PDF given by Equation 4. Probability Distributions 1.

Probability distributions are typically defined in terms of the probability density function. However, there are a number of probability functions used in applications.

For a continuous function, the probability density function pdf is the probability that the variate has the value x. Since for continuous distributions the probability at a single point is zero, this is often expressed in terms of an integral between two points.

The cumulative distribution function cdf is the probability that the variable takes a value less than or equal to x. The horizontal axis is the allowable domain for the given probability function. Since the vertical axis is a probability, it must fall between zero and one.

It increases from zero to one as we go from left to right on the horizontal axis. The percent point function ppf is the inverse of the cumulative distribution function.

For this reason, the percent point function is also commonly referred to as the inverse distribution function.