# Distribution Calculator

Cumulative distribution function (CDF), Score, Probability between two values and Probability density function (PDF)

The distribution calculator calculates the cumulative probabilities (p), the probability between two scores, and probability density for following distributions:
Normal distribution calculator, Binomial distribution calculator, T distribution calculator, F distribution calculator, Chi square distribution calculator, Poisson distribution calculator, Weibull distribution calculator, Exponential distribution calculator.

The score calculator calculates the scores (π₯β) for the following distributions:
Z score calculator, Binomial score calculator, t score calculator, F score calculator, Chi square score calculator, Poisson score calculator, Weibull score calculator, Exponential score calculator.

Choose π₯1 to calculate the cumulative probability based on the score, p(X ≤ π₯1) to calculate the score based on the cumulative probability, π₯1, π₯2 to calculate p(π₯1 ≤ X ≤ π₯2) or p(X ≤ π₯1), p(X ≤ π₯2) to calculate x1, x2, p(π₯1 ≤ X ≤ π₯2)

### What is a probability density function (PDF)?

For a continuous distribution, the density is the derivative of the cumulative distribution function.
For example, the bell curve line represents the density of the normal distribution.
The area under the density for a specific X range, the integral of the range, is the probability of getting value in this range. p(X=π₯) = 0.

### What is a probability mass function (PMF)?

For a discrete probability distribution, the density of value π₯ is the probability of getting this value: p(X=π₯).

### Normal distribution

The normal distribution calculator and z score calculator uses the normal distribution.
The normal distribution (also known as the Gaussian distribution), is the most widely used in statistical analyses.
This is generally because many natural processes are naturally distributed or have a very similar spread.
Some examples of normally distributed data include height, weight, and error in measurements.

The Normal distribution has a symmetric "Bell Curve" structure. more data exist around the center, which is the average, and as further the value is from the center the less likely it occurs.

Usually, when adding independent random variables, the result tends toward the normal distribution (CLT - The Central Limit Theorem)
You can calculate the values of any normal distribution based on the standard normal distribution (a normal distribution with mean equals zero and standard deviation equals one)
when X distributes normally, μ mean and σ standard deviation, Z=(x-μ)/σ distributes as the standard normal distribution, so you can calculate any normal distribution based on the standard normal distribution.

 PDF(π₯) = 1 exp(- (π₯ - μ)2 ) σ√(2π) 2σ2
Z - Standard distribution score - normal distribution with μ=0 and σ=1.
 Z = π₯ - μ σ

### Binomial distribution

The binomial distribution calculator and binomial score calculator uses the binomial distribution.
The binomial distribution is a discrete distribution, that calculates the probability of getting a specific number of successes in an experiment with n trials and p (probability of success).

When calculating the score (percentile), there is usually no X that meets the exact probability you enter. The tool will calculate the X that will generate a probability that is equal to or bigger than the input probability but will calculate the probabilities for both X and X-1.

When the tool can't calculate the distribution or the density using the binomial distribution, due to large sample size and/or a large number of successes, it will use the normal approximation with μ = np and σ=√(np(1-p)), or for the z-score calculation, it may be a combination between the two distributions using the binomial distribution whenever is possible.
 P(X=π₯) = ( π₯ )pπ₯qn-π₯ n
 Z = π₯ - np √(np(1 - p))

### Student's t-distribution

The t distribution calculator and t score calculator uses the student's t-distribution.
The T-student distribution is an artificial distribution used for a normally distributed population, when we don't know the population's standard deviation or when the sample size is too small.

T distribution looks similar to the normal distribution but lower in the middle and with thicker tails. The shape depends on the degrees of freedom, number of independent observations, usually number of observations minus one (n-1). The higher the degree of freedom the more it resembles the normal distribution.

 PDF = 1 (1+ π₯2 )-(k+1)/2 B(1/2,k/2)√k k

### Chi-squared distribution

The chi square distribution calculator and chi square score calculator uses the chi-squared distribution.
The chi-Square distribution is used for a normally distributed population, as an accumulation of independent squared standard normal random variables.

Let Z1, Z2, ... Zk be independent standard random variables.
Let X= [Z12+ Z22+....+Zk2].
X distributes as a Chi-square random variable with k degrees of freedom.

 PDF(π₯, k) = 1 xk/2-1e-π₯/2 2k/2Γ(k/2)

### F distribution

The F distribution calculator and F score calculator uses the FisherβSnedecor distribution.
The F (Fisher Snedecor) distribution is used for a normally distributed population. as a division accumulation of independent squared standard normal random variables, or division between two chi-squared variables.

Let Z1, Z2,....Zn be independent standard random variables.
Let X1 = [Z12 + Z22 +....+ Zn2].

Let Z'1, Z'2,....Z'n be independent standard random variables.
Let X2 = [Z'12 + Z'22 +....+ Z'm2].

 Let X = X1/n X2/m
X distributes as F random variable with n degrees of freedom (numerator) and m degrees of freedom (denominator)

X1 distribute as a chi-square random variable with n degrees of freedom.
X2 distribute as a chi-square random variable with m degrees of freedom.

Example of use: ANOVA test, F test for variances comparison.

### Poisson distribution

The poisson distribution calculator and poisson score calculator uses the poisson distribution.
The Poisson distribution is a discrete distribution that describes the probability of getting the number of events in a fixed unit of time. All the events are independent.
λ is the average number of events per unit of time.
The number of events on t units of time distributes Poisson with average number of events.
The time between events distributes Exponential with mean equals 1/λ.

π₯ ≥ 0

 P(X=π₯) = λxe-λ π₯!

### Exponential distribution

The exponential distribution calculator and exponential score calculator uses the exponential distribution.
The Exponential distribution is the complementary distribution for the Poisson distribution, it represents the distribution of the time between events.
A unique character of the distribution is memorylessness - the distribution of the time from now to the next event does not depend on the time you already waited.
Memorylessness example: If the probability for a burned-out bulb event in the next 2 months is 0.3, and you waited 1 year without any event, now the probability for an event in the next 2 months is still 0.3.
λ - duration between the events
When using λ be sure to check if it is the duration between events or rate - events per unit of time since some people use λ as duration and some use λ as rate: rate = 1 / duration.

π₯ ≥ 0
 P(X=π₯) = e-π₯/λ λ
P(X≤π₯) = 1 - e-π₯/λ

Example: When the event is a faulty lamp, and the average number of lamps that need to be replaced in a month is 16.
The number of lamps that need to be replaced in 5 months distributes Pois(80). since: 5 * 16 = 80.
The time between faulty lamp evets distributes Exp(1/16). The unit is months.

### Weibull distribution

The Weibull distribution calculator and Weibull score calculator uses the weibull distribution.
The Weibull distribution is a continuous distribution that is used for reliability as a lifetime distribution.
When k = 1 (shape), the failure rate is constant. This is the exponential distribution.
When k > 1 (shape), the failure rate increases over time.