信貸利率試算公式 相關
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Pearson correlation coefficient. Several sets of ( x , y) points, with the correlation coefficient of x and y for each set. The correlation reflects the strength and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom).
Central limit theorem. In probability theory, the central limit theorem ( CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed.
Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation. In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable.
- Definition
- Properties
- Parameter Estimation
- Applications
- Related Distributions
- See Also
- Bibliography
- External Links
Standard parameterization
The probability density function of a Weibull random variableis 1. f ( x ; λ , k ) = { k λ ( x λ ) k − 1 e − ( x / λ ) k , x ≥ 0 , 0 , x < 0 , {\displaystyle f(x;\lambda ,k)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0,\end{cases}}} where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distr...
Density function
The form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k < 1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing. For k = 1, the density function tends to 1/λ as x approaches zero from above and is strictly decreasing. For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at x =...
Cumulative distribution function
The cumulative distribution functionfor the Weibull distribution is 1. F ( x ; k , λ ) = 1 − e − ( x / λ ) k {\displaystyle F(x;k,\lambda )=1-e^{-(x/\lambda )^{k}}\,} for x ≥ 0, and F(x; k; λ) = 0 for x< 0. If x = λ then F(x; k; λ) = 1 − e−1 ≈ 0.632 for all values of k. Vice versa: at F(x; k; λ) = 0.632 the value of x ≈ λ. The quantile (inverse cumulative distribution) function for the Weibull distribution is 1. Q ( p ; k , λ ) = λ ( − ln ( 1 − p ) ) 1 / k {\displaystyle Q(p;k,\lambda )=\la...
Moments
The moment generating function of the logarithm of a Weibull distributed random variableis given by 1. E [ e t log X ] = λ t Γ ( t k + 1 ) {\displaystyle \operatorname {E} \left[e^{t\log X}\right]=\lambda ^{t}\Gamma \left({\frac {t}{k}}+1\right)} where Γ is the gamma function. Similarly, the characteristic function of log Xis given by 1. E [ e i t log X ] = λ i t Γ ( i t k + 1 ) . {\displaystyle \operatorname {E} \left[e^{it\log X}\right]=\lambda ^{it}\Gamma \left({\frac {it}{k}}+1\ri...
Ordinary least square using Weibull plot
The fit of a Weibull distribution to data can be visually assessed using a Weibull plot. The Weibull plot is a plot of the empirical cumulative distribution function F ^ ( x ) {\displaystyle {\widehat {F}}(x)} of data on special axes in a type of Q–Q plot. The axes are ln ( − ln ( 1 − F ^ ( x ) ) ) {\displaystyle \ln(-\ln(1-{\widehat {F}}(x)))} versus ln ( x ) {\displaystyle \ln(x)} . The reason for this change of variables is the cumulative distribution function can be linearized: 1. F...
Method of moments
The coefficient of variationof Weibull distribution depends only on the shape parameter: 1. C V 2 = σ 2 μ 2 = Γ ( 1 + 2 k ) − ( Γ ( 1 + 1 k ) ) 2 ( Γ ( 1 + 1 k ) ) 2 . {\displaystyle CV^{2}={\frac {\sigma ^{2}}{\mu ^{2}}}={\frac {\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}}{\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}}}.} Equating the sample quantities s 2 / x ¯ 2 {\displaystyle s^{2}/{\bar {x}}^{2}} to σ 2 / μ 2 {\displaystyle \sigma ^{...
Maximum likelihood
The maximum likelihood estimator for the λ {\displaystyle \lambda } parameter given k {\displaystyle k} is 1. λ ^ = ( 1 n ∑ i = 1 n x i k ) 1 k {\displaystyle {\widehat {\lambda }}=({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{k})^{\frac {1}{k}}} The maximum likelihood estimator for k {\displaystyle k} is the solution for kof the following equation 1. 0 = ∑ i = 1 n x i k ln x i ∑ i = 1 n x i k − 1 k − 1 n ∑ i = 1 n ln x i {\displaystyle 0={\frac {\sum _{i=1}^{n}x_{i}^{k}\ln x_{i}}{\sum _{i=1}^{n}x...
The Weibull distribution is used[citation needed] 1. In survival analysis 2. In reliability engineering and failure analysis 3. In electrical engineeringto represent overvoltage occurring in an electrical system 4. In industrial engineering to represent manufacturing and deliverytimes 5. In extreme value theory 6. In weather forecasting and the win...
If W ∼ W e i b u l l ( λ , k ) {\displaystyle W\sim \mathrm {Weibull} (\lambda ,k)} , then the variable G = log W {\displaystyle G=\log W} is Gumbel (minimum) distributed with location parameter...A Weibull distribution is a generalized gamma distribution with both shape parameters equal to k.The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter. It has the probability density function f ( x ; k , λ , θ ) = k λ ( x − θ λ ) k − 1 e − ( x − θ λ ) k...The Weibull distribution can be characterized as the distribution of a random variable W {\displaystyle W} such that the random variable X = ( W λ ) k {\displaystyle X=\left({\frac {W}{\lambda }}\r...Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (...Mann, Nancy R.; Schafer, Ray E.; Singpurwalla, Nozer D. (1974), Methods for Statistical Analysis of Reliability and Life Data, Wiley Series in Probability and Mathematical Statistics: Applied Proba...Muraleedharan, G.; Rao, A.D.; Kurup, P.G.; Nair, N. Unnikrishnan; Sinha, Mourani (2007), "Modified Weibull Distribution for Maximum and Significant Wave Height Simulation and Prediction", Coastal E..."Weibull distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]e. In statistics, linear regression is a statistical model which estimates the linear relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables ). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear ...
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although ...
For instance, consider a call center which receives, randomly, an average of λ = 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next one will arrive. Under these assumptions, the number k of calls received during any minute has a Poisson probability distribution ...
