Amortization (business)

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b). It is the maximum entropy probability distribution for a random variate X under no constraint other than that it is contained in the distribution's support.

Characterization

Probability density function

The probability density function of the continuous uniform distribution is:

$f(x)={\begin{cases}{\frac {1}{b-a}}&\mathrm {for} \ a\leq x\leq b,\\[8pt]0&\mathrm {for} \ x<a\ \mathrm {or} \ x>b\end{cases}}$

The values of f(x) at the two boundaries a and b are usually unimportant because they do not alter the values of the integrals of f(x) dx over any interval, nor of x f(x) dx or any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be 1/(b − a). The latter is appropriate in the context of estimation by the method of maximum likelihood. In the context of Fourier analysis, one may take the value of f(a) or f(b) to be 1/(2(b − a)), since then the inverse transform of many integral transforms of this uniform function will yield back the function itself, rather than a function which is equal "almost everywhere", i.e. except on a set of points with zero measure. Also, it is consistent with the sign function which has no such ambiguity.

In terms of mean μ and variance σ², the probability density may be written as:

$f(x)={\begin{cases}{\frac {1}{2\sigma {\sqrt {3}}}}&{\mbox{for }}-\sigma {\sqrt {3}}\leq x-\mu \leq \sigma {\sqrt {3}}\\0&{\text{otherwise}}\end{cases}}$

Cumulative distribution function

The cumulative distribution function is:

$F(x)={\begin{cases}0&{\text{for }}x<a\\[8pt]{\frac {x-a}{b-a}}&{\text{for }}a\leq x<b\\[8pt]1&{\text{for }}x\geq b\end{cases}}$

Its inverse is:

$F^{-1}(p)=a+p(b-a)\,\,{\text{ for }}0<p<1$

In mean and variance notation, the cumulative distribution function is:

$F(x)={\begin{cases}0&{\text{for }}x-\mu <-\sigma {\sqrt {3}}\\{\frac {1}{2}}\left({\frac {x-\mu }{\sigma {\sqrt {3}}}}+1\right)&{\text{for }}-\sigma {\sqrt {3}}\leq x-\mu <\sigma {\sqrt {3}}\\1&{\text{for }}x-\mu \geq \sigma {\sqrt {3}}\end{cases}}$

and the inverse is:

$F^{-1}(p)=\sigma {\sqrt {3}}(2p-1)+\mu \,\,{\text{ for }}0\leq p\leq 1$

Generating functions

Moment-generating function

The moment-generating function is:

$M_{x}=E(e^{tx})={\frac {e^{tb}-e^{ta}}{t(b-a)}}\,\!$

from which we may calculate the raw moments m _k

$m_{1}={\frac {a+b}{2}},\,\!$

$m_{2}={\frac {a^{2}+ab+b^{2}}{3}},\,\!$

$m_{k}={\frac {1}{k+1}}\sum _{i=0}^{k}a^{i}b^{k-i}.\,\!$

For a random variable following this distribution, the expected value is then m₁ = (a + b)/2 and the variance is m₂ − m₁² = (b − a)²/12.

Cumulant-generating function

For n ≥ 2, the nth cumulant of the uniform distribution on the interval [0, 1] is b_{n/n, where bn is the nth Bernoulli number.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.}

Properties

Moments and parameters

The first two moments of the distribution are:

$E(X)={\frac {1}{2}}(a+b)$

$V(X)={\frac {1}{12}}(b-a)^{2}$

Solving these two equations for parameters a and b, given known moments E(X) and V(X), yields:

$a=E(X)-{\sqrt {3V(X)}}$

$b=E(X)+{\sqrt {3V(X)}}$

Order statistics

Let X₁, ..., X_n be an i.i.d. sample from U(0,1). Let X_(k) be the kth order statistic from this sample. Then the probability distribution of X_(k) is a Beta distribution with parameters k and n − k + 1. The expected value is

$\operatorname {E} (X_{(k)})={k \over n+1}.$

This fact is useful when making Q-Q plots.

The variances are

$\operatorname {Var} (X_{(k)})={k(n-k+1) \over (n+1)^{2}(n+2)}.$

Uniformity

The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself (but it is dependent on the interval size), so long as the interval is contained in the distribution's support.

To see this, if X ~ U(a,b) and [x, x+d] is a subinterval of [a,b] with fixed d > 0, then

$P\left(X\in \left[x,x+d\right]\right)=\int _{x}^{x+d}{\frac {\mathrm {d} y}{b-a}}\,={\frac {d}{b-a}}\,\!$

which is independent of x. This fact motivates the distribution's name.

Generalization to Borel sets

This distribution can be generalized to more complicated sets than intervals. If S is a Borel set of positive, finite measure, the uniform probability distribution on S can be specified by defining the pdf to be zero outside S and constantly equal to 1/K on S, where K is the Lebesgue measure of S.

Standard uniform

Restricting $a=0$ and $b=1$, the resulting distribution U(0,1) is called a standard uniform distribution.

One interesting property of the standard uniform distribution is that if u₁ has a standard uniform distribution, then so does 1-u₁. This property can be used for generating antithetic variates, among other things.

Related distributions

• If X has a standard uniform distribution, then by the inverse transform sampling method, Y = − λ ln(X) has an exponential distribution with (rate) parameter λ.
• If X has a standard uniform distribution, then Y = Xⁿ has a beta distribution with parameters 1/n and 1. (Note this implies that the standard uniform distribution is a special case of the beta distribution, with parameters 1 and 1.)
• The Irwin–Hall distribution is the sum of n i.i.d. U(0,1) distributions.
• The sum of two independent, equally distributed, uniform distributions yields a symmetric triangular distribution.
• The distance between two i.i.d. uniform random variables also has a triangular distribution, although not symmetric.

Relationship to other functions

As long as the same conventions are followed at the transition points, the probability density function may also be expressed in terms of the Heaviside step function:

$f(x)={\frac {\operatorname {H} (x-a)-\operatorname {H} (x-b)}{b-a}},\,\!$

or in terms of the rectangle function

$f(x)={\frac {1}{b-a}}\,\operatorname {rect} \left({\frac {x-\left({\frac {a+b}{2}}\right)}{b-a}}\right).$

There is no ambiguity at the transition point of the sign function. Using the half-maximum convention at the transition points, the uniform distribution may be expressed in terms of the sign function as:

$f(x)={\frac {\operatorname {sgn} {(x-a)}-\operatorname {sgn} {(x-b)}}{2(b-a)}}.$

Applications

In statistics, when a p-value is used as a test statistic for a simple null hypothesis, and the distribution of the test statistic is continuous, then the p-value is uniformly distributed between 0 and 1 if the null hypothesis is true.

Sampling from a uniform distribution

There are many applications in which it is useful to run simulation experiments. Many programming languages have the ability to generate pseudo-random numbers which are effectively distributed according to the standard uniform distribution.

If u is a value sampled from the standard uniform distribution, then the value a + (b − a)u follows the uniform distribution parametrised by a and b, as described above.

Sampling from an arbitrary distribution

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

The uniform distribution is useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable. This method is very useful in theoretical work. Since simulations using this method require inverting the CDF of the target variable, alternative methods have been devised for the cases where the cdf is not known in closed form. One such method is rejection sampling.

The normal distribution is an important example where the inverse transform method is not efficient. However, there is an exact method, the Box–Muller transformation, which uses the inverse transform to convert two independent uniform random variables into two independent normally distributed random variables.

Quantization error

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. In analog-to-digital conversion a quantization error occurs. This error is either due to rounding or truncation. When the original signal is much larger than one least significant bit (LSB), the quantization error is not significantly correlated with the signal, and has an approximately uniform distribution. The RMS error therefore follows from the variance of this distribution.

Estimation

Estimation of maximum

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Given a uniform distribution on [0, N] with unknown N, the UMVU estimator for the maximum is given by

${\hat {N}}={\frac {k+1}{k}}m=m+{\frac {m}{k}}$

where m is the sample maximum and k is the sample size, sampling without replacement (though this distinction almost surely makes no difference for a continuous distribution). This follows for the same reasons as estimation for the discrete distribution, and can be seen as a very simple case of maximum spacing estimation. This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II.

Estimation of midpoint

The midpoint of the distribution (a + b) / 2 is both the mean and the median of the uniform distribution. Although both the sample mean and the sample median are unbiased estimators of the midpoint, neither is as efficient as the sample mid-range, i.e. the arithmetic mean of the sample maximum and the sample minimum, which is the UMVU estimator of the midpoint (and also the maximum likelihood estimate).

Confidence interval for the maximum

Let X₁, X₂, X₃, ..., X_n be a sample from U( 0, L ) where L is the population maximum. Then X_(n) = max( X₁, X₂, X₃, ..., X_n ) has the density

$f_{n}(X_{(n)})=n{\frac {1}{L}}\left({\frac {X_{(n)}}{L}}\right)^{n-1}=n{\frac {X_{(n)}^{n-1}}{L^{n}}},0<X_{n}<L$

The confidence interval for the estimated population maximum is then ( X_(n), X_(n) / α^1/n ) where 100 ( 1 - α )% is the confidence level sought. In symbols

$X_{(n)}\leq L\leq X_{(n)}/\alpha ^{1/n}$

See also

• Beta distribution
• Box–Muller transform
• Probability plot
• Q-Q plot
• Rectangular function
• Uniform distribution (discrete)

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Online calculator of Uniform distribution (continuous)

55 yrs old Metal Polisher Records from Gypsumville, has interests which include owning an antique car, summoners war hack and spelunkering. Gets immense motivation from life by going to places such as Villa Adriana (Tivoli).

my web site - summoners war hack no survey ios Template:Common univariate probability distributions hu:Egyenletes eloszlás su:Sebaran seragam#Kasus kontinyu