Uniform Distribution
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A random variable X has a continuous uniform distribution if its probability density function f(x) is constant over the interval (α, β). The uniform distribution has a probability density function f(x) = k for α < x < β, where k is a constant, and is equal to 0 otherwise. All values from α to β are equally likely to occur, meaning the probability of X falling in any sub-interval of (α, β) is equal regardless of the interval's position within the range.
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3. The chi-squared distribution, which is a special case of the gamma distribution where the shape parameter equals degrees of freedom. It is used to model outcomes of summing squared standard normal random variables.
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Cochran's Q test is a nonparametric test used to compare dependent categorical variables across three or more conditions. It is an extension of the McNemar test for related samples. The test compares the probability of a target response across different conditions to see if treatments have identical effects. It assumes a large sample size and that blocks were randomly selected. An example uses data from pet stores to test if they are equally likely to display both reptiles (snakes and lizards) during different times of the year. The Cochran's Q test rejects the null hypothesis, finding stores were more likely to display both during Christmas than other times.
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1. Continuous random variables are defined over intervals rather than discrete points. The probability that a continuous random variable takes on a value in an interval from a to b is given by an integral of the probability density function f(x) over that interval.
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This document provides an overview of probability distributions and related concepts. It defines key probability distributions like the binomial, beta, multinomial, and Dirichlet distributions. It also describes probability distribution equations like the cumulative distribution function and probability density function. Additionally, it outlines descriptive parameters for distributions like mean, variance, skewness and kurtosis. Finally, it briefly discusses probability theorems such as the law of large numbers, central limit theorem, and Bayes' theorem.
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A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
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This document discusses common probability distributions used in hypothesis testing. It describes discrete and continuous random variables and their probability distributions. Specifically, it covers the discrete uniform distribution, binomial distribution, and normal distribution. The normal distribution is particularly important as it is widely used in quantitative analysis due to the central limit theorem stating that sums of independent random variables are approximately normally distributed.
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Simulation
Simulation involves imitating the performance of a system using a model to generate sample experiments without observing the real system. Monte Carlo simulation specifically involves an element of chance and is useful when direct experimentation is impossible or too costly. It generates random numbers that can represent values of random variables to simulate observations. Common techniques include using uniform random numbers between 0 and 1 to represent distributions and the Box-Muller method for generating normal random variables.
Sampling Distributions
The sampling distribution of the mean describes the probability distribution of sample means that would be obtained by drawing all possible random samples of a given size from a population. For a random sample of size n drawn from a normal population with mean μ and standard deviation σ, the sampling distribution of the mean is a normal distribution with mean μ and standard deviation σ/√n. When the population standard deviation σ is unknown, the t distribution provides the sampling distribution of the mean if the population is normal. The t distribution approaches the normal distribution as the degrees of freedom increase.
Interval Estimation & Estimation Of Proportion
This document discusses interval estimation and estimation of proportions from sample data. It defines an interval estimate as a range of values (L1 to L2) that is likely to contain the true population parameter. A 100(1-α)% confidence interval is an interval where the probability of the parameter falling within the interval is 1-α. For large samples where the population variance is known, the confidence interval for the mean is calculated. For small samples or unknown variance, approximations are used. Estimation of a population proportion p is discussed, where the sample proportion p=X/n is an unbiased estimator of p. For large samples, a normal approximation can be used to construct a confidence interval for p. Sample size determination is
Point Estimation
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Normal Distribution
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is sufficiently large.
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The document discusses several probability distributions:
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2) The Poisson process, which assumes events occur continuously with a constant average rate λ and the time between events is exponentially distributed.
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Hypergeometric Distribution
The hypergeometric distribution models sampling without replacement from a finite population. It gives the probability of getting x successes in n draws from a population of size N that contains a number of successes. The mean is equal to n(a/N) and the variance is equal to n(a/N)(1-a/N)(N-n)/(N-1). When the sample size n is small compared to the population N, the binomial distribution is a good approximation to the hypergeometric.
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Mathematical Expectation And Variance
1. Mathematical expectation, or expected value, of a random variable is calculated by taking the sum of each possible outcome value multiplied by its probability of occurring.
2. For a game with multiple outcomes and their associated probabilities and payoffs, the expected value is the average amount a player can expect to win or lose over many plays. A positive expected value means the game favors the player, while a negative value means it is biased against the player.
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The document discusses the classical definition of probability as well as axioms that define probability mathematically. It introduces the classical definition where probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. It then discusses limitations of the classical definition and introduces the frequency interpretation of probability. Finally, it outlines three axioms that define a function as a valid probability function: 1) probabilities are between 0 and 1, 2) the total probability of the sample space is 1, and 3) probabilities of mutually exclusive events sum to the total probability.
Sample Space And Events
1) A sample space is the set of all possible outcomes of an experiment. Any subset of the sample space is called an event.
2) Events can be mutually exclusive, meaning they cannot occur simultaneously, or they can overlap. The relationships between events can be represented using Venn diagrams.
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Uniform Distribution
- 2. Rectangular or Uniform distribution A random variable X is said to have a continuous uniform distribution over an interval (, ) if its probability density function is constant k over entire range of x. PROBABILITY DENSITY FUNCTION f (x) = k, < X < = 0 otherwise
- 3. Rectangular or Uniform distribution The uniform distribution, with parameters and , has probability density function
- 4. Figure:Graph of uniform probability density All values of x from to are equally likely in the sense that the probability that x lies in an interval of width x entirely contained in the interval from to is equal to x/( - ), regardless of the exact location of the interval. Uniform distribution
- 5. Distribution function for uniform density function Uniform distribution
- 6. The Uniform Distribution Mean of uniform distribution Proof:
- 7. The Uniform Distribution Variance of uniform distribution Proof:
- 8. The Uniform Distribution Moment generating function
- 9. Discrete Uniform distribution If random variable assume finite no. of values with each value occuring with same probability Probability density function is f(x) = 1/n, X=x1,x2,…… xn