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 formulae
3 answers
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.
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.