# Distribution Calculator

This distribution calculator determines the Cumulative Distribution Function (CDF), scores, probabilities between two scores, and PDF or PMF for the following distributions: Normal, Binomial, Student's t, F, Chi-Square, Poisson, Weibull, Exponential, and Uniform.

The Distribution Calculator computes Cumulative Probabilities (p), Probabilities between two scores, and Probability Density for the 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
• Uniform Distribution Calculator
The Score Calculator computes Scores (𝑥₁) 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
• Uniform Score Calculator
Use the calculator by specifying:
• 𝑥1 to calculate the Cumulative Probability based on the Score
• p(X ≤ 𝑥1) to calculate the Score based on the Cumulative Probability
• 𝑥1, 𝑥2 to calculate p(𝑥1 ≤ X ≤ 𝑥2)
• p(X ≤ 𝑥1), p(X ≤ 𝑥2) to calculate 𝑥1, 𝑥2, and p(𝑥1 ≤ X ≤ 𝑥2)

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

The Probability Density Function (PDF), indicated as f(x), is relevant to a continuous random variable. It explains the relative likelihood of a given value. It means that you would expect to encounter more values around a higher PDF than around a lower PDF. To calculate the probability that a random variable will fall in a specific range, you need to calculate the area under the density curve. You may do it by calculating the integral of the density. The probability of obtaining an exact value of a continuous random variable is zero.

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

The Probability Mass Function (PMF) applies to discrete probability distributions. The PMF represents the probability of obtaining a specific score within the distribution. The PMF and the PDF represent likelihood, one for a discrete distribution and one for a continuous distribution.

### Normal distribution calculator

The normal distribution calculator and z-score calculator use the normal distribution. The normal distribution is also known as the Gaussian distribution. The normal distribution is the most utilized in statistical analysis. This is because many natural processes follow the normal distribution, or resemble it For example, the height, weight, and measurement error follow the normal distribution.
The Normal distribution has a symmetrical "Bell Curve" shape. Most of the data centers around the average, and the frequency decreases as values move farther away from the center.

According to the Central Limit Theorem (CLT), the distribution of the sum or average of independent random variables tends to approach a normal distribution as the number of variables increases
This means that the distribution of the average or sum of non-normal distributions may look like the normal distribution.
The larger the sample size, the more the distribution resembles the normal distribution.

You may calculate values for any normal distribution, using the standard normal distribution. The standard normal distribution is a special case of the normal distribution. It contains the following parameters: a mean of 0 and a standard deviation of 1.
When X follows a normal distribution with mean (μ) and standard deviation (σ), the standardized value Z = (x-μ)/σ follows the standard normal distribution. As a result, values for any normal distribution can be calculated based on the standard normal distribution.

 PDF(𝑥) = f(x) = 1 exp(- (𝑥 - μ)2 ) σ√(2π) 2σ2

Z - Standard distribution score - normal distribution with μ=0 and σ=1.

 Z = 𝑥 - μ σ

### Binomial distribution calculator

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=x) = ( x )pxqn-x n
 Z = x - np √(np(1 - p))

### Student's t-distribution calculator

The t distribution calculator and t score calculator uses the student's t-distribution.
The Student's t-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+ x2 )-(k+1)/2 B(1/2,k/2)√k k

### Chi-squared distribution calculator

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(x, k) = 1 xk/2-1e-x/2 2k/2Γ(k/2)

### F distribution calculator

The F distribution calculator and F score calculator uses the Fisher–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.

Consider n independent standard normal random variables: Z1, Z2,....Zn.
Let X1 = [Z12 + Z22 +....+ Zn2].
X1 follows a chi-square distribution with n degrees of freedom.

Also, consider m independent standard normal random variables: Z'1, Z'2,....Z'm.
Let X2 = [Z'12 + Z'22 +....+ Z'm2].
X2 follows a chi-square distribution with m degrees of freedom.

">Let X =X1/nX2/m

X follows an F distribution with n degrees of freedom in the numerator, and m degrees of freedom in the denominator.

Applications of the F distribution include the ANOVA test and the F test for comparing variances.

### Poisson distribution calculator

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 t*λ average number of events.
The time between events distributes Exponential with mean equals 1/λ.

x ≥ 0

 P(X=x) = λxe-λ x!

### Exponential distribution calculator

The Exponential Distribution Calculator and the Exponential Score Calculator utilize the Exponential Distribution.
The Exponential Distribution is the complementary distribution of the Poisson Distribution, and models the time between events. One key feature of the distribution is its memorylessness, meaning the distribution of time from the present to the next event is not influenced by the time already elapsed.
The concept of memorylessness in the exponential distribution is illustrated by the example of a burned-out bulb. If the probability of a bulb burning out in the next 2 months is 0.3, this probability remains the same even if the bulb has already lasted for 1 year without burning out.
It's important to note that the value of λ in exponential distribution can represent either the duration between events or the rate of events per unit of time. It's crucial to determine which definition is being used, as some people use λ to represent the duration, while others use it to represent the rate. To convert between the two, simply remember that the rate is equal to 1 divided by the duration.

In the following formula, λ represents the duration between the events

x ≥ 0

 P(X=x) = e-x/λ λ

P(X≤x) = 1 - e-x/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 calculator

The Weibull Distribution Calculator and Weibull Score Calculator use the Weibull Distribution, a continuous probability distribution commonly employed in reliability analysis as a lifetime distribution. When the shape parameter (k) is equal to 1, the failure rate is constant, resulting in an exponential distribution. On the other hand, if k is greater than 1, the failure rate increases with time.

### Uniform distribution calculator

The Uniform Distribution Calculator and Uniform Score Calculator utilize the the Uniform Distribution, , a continuous probability distribution with equal probability density along the distribution range.
Min (also known as 'a') - the minimum possible value.
Max (also known as 'b') - the maximum possible value.

##### Uniform distribution formula
 PDF(x) = f(x) = 1 for Min ≤ x ≤Max Max - Min PDF(x) = f(x) = 0 for x < Min or x > Max
 F(x) = 0 for x < Min or x > Max F(x) = x - Min for Min ≤ x ≤ Max Max - Min F(x) = 1 for x > Max