without the use of the standard normal tables, should techniques of calculus be used to find the probabilities concerning a normal distribution?
Why would you want to do the work again? The values on the standard normal tables were found using calculus.
Yes, techniques of calculus can be used to find probabilities concerning a normal distribution without using standard normal tables. The probability density function (PDF) of a normal distribution can be expressed as:
f(x) = (1 / σ√(2π)) * e^(-((x-μ)² / (2σ²)))
Where:
- f(x) is the probability density function at point x
- μ is the mean of the distribution
- σ is the standard deviation
To find the probability of a range of values, you can integrate the probability density function over that range. For example, to find the probability of an interval [a, b], you would evaluate the following integral:
P(a ≤ X ≤ b) = ∫(a to b) f(x) dx
Note that this integral requires calculus techniques, such as integration. However, if you have the mean (μ) and standard deviation (σ) of the normal distribution, you can directly compute the probability using calculus without referring to standard normal tables.