Distribution Calculator

The Distribution Calculator computes Cumulative Probabilities (p), Probabilities between two scores, and Probability Density for the following distributions:

• Normal Distribution
• Binomial Distribution
• T Distribution
• F Distribution
• Chi Square Distribution
• Poisson Distribution
• Weibull Distribution
• Exponential Distribution

The Score Calculator computes Scores (𝑥₁) for the following distributions:

• Z Score
• Binomial Score
• T Score
• F Score
• Chi Square Score
• Poisson Score
• Weibull Score
• Exponential Score

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) is the derivative of the Cumulative Distribution Function (CDF) of a continuous distribution, and describes the relative likelihood of a given value of the random variable. The area under the density curve within a specific range of X, obtained by taking the integral of the range, represents the probability of getting a value in that range. Since the distribution is continuous, the probability of getting an exact value is zero. For example, the bell curve is a graphical representation of the density of a normal distribution.

What is a probability mass function (PMF)?

The probability mass function is applicable to discrete probability distributions and is equivalent to the Probability Density Function for continuous distributions. The PMF represents the probability of obtaining a specific score in the distribution."

Normal distribution

The normal distribution calculator and z-score calculator utilize the normal distribution, also known as the Gaussian distribution, which is the most widely used in statistical analysis. This is due to the fact that many natural processes exhibit a natural distribution or closely resemble a normal distribution in their spread. Examples of data that commonly follow a normal distribution include height, weight, and measurement error.
The Normal distribution is characterized by its symmetrical "Bell Curve" shape, with the majority of data concentrated around the average (center), and a decrease in frequency as values deviate further 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

Values for any normal distribution can be calculated using the standard normal distribution (mean of 0 and standard deviation of 1) as a reference.
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) =	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

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

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

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, ... Z_k be independent standard random variables.
Let X= [Z₁²+ Z₂²+....+Z_k²].
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

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,....Z_n.
Let X₁ = [Z₁² + Z₂² +....+ Z_n²].
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/n
	X2/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

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 tλ 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

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

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.