Question

Gaussians, an important function in statistics, is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²). What is the standard deviation of this Gaussian distribution?

a) 1/a
b) √a
c) 1/√a
d) a²

Answer 1

The standard deviation of the Gaussian distribution given by the function f(x) = e^(-ax²) is 1/√a.

Step-by-step explanation:

The standard deviation of the Gaussian distribution given by the function f(x) = e^(-ax²) is 1/√a.

To find the standard deviation, we can equate it to the square root of the variance. The variance of a continuous probability distribution is given by the formula o² = Σ (x − µ)² P(x), where x represents the values, µ is the mean, and P(x) represents the corresponding probability. For the Gaussian distribution, we have f(x) = e^(-ax²), and the mean µ is zero since the curve is symmetrical.

Therefore, the variance o² is given by:

o² = ∫ (x - 0)² e^(-ax²) dx

Using integration techniques, we can solve this integral to find the variance. Then, taking the square root of the variance gives us the standard deviation:

o = √(o²)

Hence, the standard deviation is 1/√a.