Normal distribution formulae
The formulas for the normal distribution are as follows:
1. Probability Density Function (PDF):
The probability density function of the normal distribution is given by the formula:
f(x) = (1 / sqrt(2 * π * σ^2)) * e^(-((x - μ)^2) / (2 * σ^2))
where:
- f(x) is the probability density function at x
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
- e is the base of the natural logarithm (approximately equal to 2.71828)
2. Cumulative Distribution Function (CDF):
The cumulative distribution function of the normal distribution is given by the formula:
F(x) = (1 / 2) * (1 + erf((x - μ) / (σ * sqrt(2))))
where:
- F(x) is the cumulative distribution function at x
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
- erf is the error function
These formulas can be used to calculate probabilities and perform statistical calculations related to the normal distribution.
Normal distribution formula integral formula
The integral formula for the normal distribution is used to calculate probabilities within a certain range of values. It is given by the cumulative distribution function (CDF) of the normal distribution:
F(x) = ∫[μ, x] (1 / sqrt(2 * π * σ^2)) * e^(-((t - μ)^2) / (2 * σ^2)) dt
where:
- F(x) is the cumulative distribution function at x
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
- t is the variable of integration
- ∫[μ, x] denotes integration from the mean (μ) to the value x
This formula allows you to calculate the probability of a random variable falling within a certain range, represented by the integration limits μ and x. By evaluating the integral, you can find the area under the probability density curve between these limits, which corresponds to the probability of the variable falling within that range.
The formula for the probability density function (PDF) of a normal distribution is given by:
f(x) = (1 / (σ * √(2π))) * e^(-(x-μ)^2 / (2 * σ^2))
Where:
- f(x) is the value of the PDF at a particular point x,
- σ is the standard deviation of the distribution,
- μ is the mean of the distribution, and
- e is the base of the natural logarithm.
The formula for the cumulative distribution function (CDF) of a normal distribution is given by:
F(x) = (1/2) * [1 + erf((x-μ) / (σ * √2))]
Where:
- F(x) is the value of the CDF up to a particular point x, and
- erf is the error function, which is a special function in mathematics.
These two formulas are commonly used to describe and calculate probabilities in a normal distribution.
The Normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is widely used in statistics. It is defined by its probability density function (PDF), which takes the form:
f(x) = (1 / (σ√(2π))) * e^(-(x-μ)^2 / (2σ^2))
Where:
- f(x) is the probability density function at a given value of x.
- μ is the mean (average) of the distribution.
- σ is the standard deviation, which measures the spread of the distribution.
- e is the base of natural logarithms, approximately equal to 2.71828.
To calculate probabilities or values related to the Normal distribution, you can use different formulas depending on the information you have.
1. Probability Density Function (PDF):
To find the probability density f(x) at a specific value of x, substitute the value into the formula mentioned above.
2. Cumulative Distribution Function (CDF):
To find the probability that a random variable X is less than or equal to a given value x, you need to calculate the cumulative distribution function. The formula for the CDF is the integral of the PDF from negative infinity to x. However, this integral does not have a simple closed-form solution, so it is commonly looked up in tables or computed using software.
3. Inverse Cumulative Distribution Function (Quantile Function):
To find the value x for which the cumulative distribution function is equal to a given probability p, you need to calculate the inverse cumulative distribution function. This is often denoted as F^(-1)(p). Similar to the CDF, this function is best obtained using tables or software.
It's worth mentioning that statistical software, calculators, or programming languages (such as Excel, Python, or R) typically provide built-in functions to compute probabilities and values related to the Normal distribution.