Write in terms of the cdf of the standard Gaussian variable with a positive argument. In other words, your answer be in terms of of where is a function of which takes only positive values for .
To write the probability density function (pdf) in terms of the cumulative distribution function (CDF) of a standard Gaussian variable with a positive argument, we can use the fact that the pdf is the derivative of the CDF.
Let's denote the CDF of the standard Gaussian variable as Φ(x), where x is the argument. Then, the derivative of Φ(x) with respect to x will give us the pdf.
Mathematically, we can express this relationship as:
f(x) = dΦ(x) / dx
For a standard Gaussian variable, the CDF can be written as:
Φ(x) = ∫[from -∞ to x] (1/√(2π)) * e^(-t^2/2) dt
Note: The integral is evaluated from negative infinity (-∞) up to the value of x.
Taking the derivative of Φ(x) with respect to x will give us the pdf:
f(x) = dΦ(x) / dx = (1/√(2π)) * e^(-x^2/2)
Therefore, the pdf of a standard Gaussian variable with a positive argument can be expressed in terms of the CDF as:
f(x) = (1/√(2π)) * e^(-x^2/2)
Where f(x) is the probability density function and Φ(x) is the cumulative distribution function.