# Normal Distribution

#### Functionalities

Draw a Mathematical-based Normal Distribution

• Draw a Normal Distribution with N(μ, σ); μ indicates the mean (location), and σ indicates its standard deviation (shape).
• Calculate the position x0 of a user-defined probability Pr(X ≤ x0) that is the possibility of a variable X being in an interval (-∞, x0] from the probability distribution.
In the curve, the left area to the red-line indicates this possibility value, and the intersection of the red line and horizontal axis (X-axis) is the x0.
• Calculate the probability Pr(μ – n × σ < X ≤ μ + n × σ) that is the possibility of a variable X being in an interval (μ – n × σ, μ + n × σ]
In the curve, the blue area indicates this possibility value.

Draw a Simulated-based Normal Distribution

• Generate and download random numbers of normal distribution using a user-defined sample size.
• Draw histogram of the generated random numbers.
• Calculate the Mean(μ) and Standard Deviation(σ) of the generated random numbers.
• Calculate the position x0 of a user-defined probability Pr(X ≤ x0) that is the possibility of a variable X being in an interval (-∞, x0] from the probability distribution of the generated random numbers.

Draw a User Data-based Normal Distribution

• Draw histogram and density plots of your data.
• Calculate the Mean(μ) and Standard Deviation(σ) of your data.
• Calculate the position x0 of a user-defined probability Pr(X ≤ x0) that is the possibility of a variable X being in an interval (-∞, x0] from the probability distribution of your data.

#### Case Example

Suppose we wanted to see the shape of N(0, 1) and wanted to know 1. at which point x0 when Pr(X < x0) = 0.025, and 2. what is the probability between means +/- 1SD

#### Mathematical-based Plot

The position of the red line and the blue ares

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

Sample descriptive statistics

# Exponential Distribution

#### Functionalities

• Draw an Exponential Distribution with E(Rate); Rate indicates the rate of change
• Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
• Get the probability distribution from simulation numbers in Simulation-based tab
• Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
• Get the probability distribution of your data which can be roughly compared to E(Rate)

#### Case Example

Suppose we wanted to see the shape of E(2), and wanted to know at which point x0 when Pr(X < x0)= 0.05

#### Mathematical-based Plot

Exponential distribution plot

The position of red line

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

Sample descriptive statistics

# Gamma Distribution

#### Functionalities

• Draw a Gamma Distribution with Gamma(α, θ); α controls the shape, 1/θ controls the change of rate
• Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
• Get the probability distribution from simulation numbers in Simulation-based tab
• Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
• Get the probability distribution of your data which can be roughly compared to Gamma(α, θ)

#### Case Example

Suppose we wanted to see the shape of Gamma(9,0.5), and wanted to know at which point x0 when Pr(X < x0)= 0.05

#### Mathematical-based Plot

Gamma distribution plot

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

Sample descriptive statistics

# Beta Distribution

#### Functionalities

• Draw a Beta Distribution with Beta(α, β); α, β controls the shape
• Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
• Get the probability distribution from simulation numbers in Simulation-based tab
• Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
• Get the probability distribution of your data which can be roughly compared to Beta(α, β)

#### Case Example

Suppose we wanted to see the shape of Beta(12, 12), and wanted to know at which point x0 when Pr(X < x0)= 0.05

#### Mathematical-based Plot

Beta distribution plot

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Explanation
• Mean = α/(α+β)
• SD = sqrt(α*β/(α+β)^2(α+β+1))
Data preview

Sample descriptive statistics

# Student's T Distribution

#### Functionalities

• Draw a T Distribution with T(v); v is the degree of freedom related to your sample size and control the shape
• Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
• Get the probability distribution from simulation numbers in Simulation-based tab
• Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
• Get the probability distribution of your data which can be roughly compared to T(v)

#### Case Example

Suppose we wanted to see the shape of T(4) and wanted to know at which point x0 when Pr(X < x0)= 0.025

#### Mathematical-based Plot

T distribution plot

The blue curve is the standard normal distribution

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

Sample descriptive statistics

# Chi-Squared Distribution

#### Functionalities

• Draw a Chi-Squared Distribution with Chi(v); v is the degree of freedom related to your sample size and control the shape
• Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
• Get the probability distribution from simulation numbers in Simulation-based tab
• Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
• Get the probability distribution of your data which can be roughly compared to Chi(v)

#### Case Example

Suppose we wanted to see the shape of Chi(4), and wanted to know at which point x0 when Pr(X < x0)= 0.05

#### Mathematical-based Plot

Chi-square distribution plot

#### Simulation-based Plot

Histogram from random numbers

When the number of bins is 0, plot will use the default number of bins

Sample descriptive statistics

Explanation
• Mean = v
• SD = sqrt(2v)
Data preview

Sample descriptive statistics

# F Distribution

#### Functionalities

• Draw a F Distribution with F(df1, df2) ; df1 and df2 are the degree of freedom related to your sample size and control the shape
• Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
• Get the probability distribution from simulation numbers in Simulation-based tab
• Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
• Get the probability distribution of your data which can be roughly compared to F(df1, df2)

#### Case Example

Suppose we wanted to see the shape of F(100, 10), and wanted to know at which point x0 when Pr(X < x0)= 0.05

#### Mathematical-based Plot

F distribution plot

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview