Let X be N(0,1) find the p.d.f of Y=|X|, a distribution that is often called the half normal.
To find the probability density function (PDF) of the half normal distribution, we need to consider the transformation Y = |X|. Here's how you can derive it step by step:
Step 1: Find the cumulative distribution function (CDF) of Y.
To do this, we need to consider the values of Y for both positive and negative values of X.
For X ≥ 0, Y = X. Therefore, P(Y ≤ y) = P(X ≤ y) since both Y and X take the same values for positive X.
So, for X ≥ 0, the CDF of Y is given by:
F(y) = P(Y ≤ y) = P(X ≤ y)
For X < 0, Y = -X. Therefore, Y becomes positive, but its sign changes.
So, for X < 0, the CDF of Y is given by:
F(y) = P(Y ≤ y) = P(-X ≤ y) = P(X ≥ -y) = 1 - P(X ≤ -y)
Step 2: Calculate the PDF of Y.
The PDF is the derivative of the CDF. Since we have defined the CDF for both positive and negative values of X, we need to express the PDF in two parts, one for X ≥ 0 and one for X < 0.
For X ≥ 0:
f(y) = dF(y)/dy = dP(X ≤ y)/dy
= d/dy ∫[0,y] (1/√(2π))e^(-t^2/2) dt (using the standard normal PDF)
= (1/√(2π))e^(-y^2/2) (simplifying the integral)
For X < 0:
f(y) = dF(y)/dy = d(1 - P(X ≤ -y))/dy
= - dP(X ≤ -y)/dy
= -d/dy ∫[-∞,-y] (1/√(2π))e^(-t^2/2) dt
= -(1/√(2π))e^(-y^2/2) (simplifying the integral)
Step 3: Combine the two parts.
To obtain the full PDF, we need to consider the positive and negative parts together. Since the distribution is symmetric, the two parts are equal in magnitude.
Therefore, the PDF of the half normal distribution is:
f(y) = (1/√(2π))e^(-y^2/2) for y ≥ 0
Note: The half normal distribution is defined for non-negative values of Y, so the PDF is only provided for y ≥ 0.