If X and Y are independent standard normal random variables, find the probability density function of Z=X/Y.
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Step-by-Step Solution
Step 1: Define the joint PDF of X and Y
Since X and Y are independent standard normal random variables, their joint probability density function (PDF) is the product of their individual PDFs. The PDF of a standard normal variable is given by f(t)=2π1e−t2/2.
Step 2: Use the transformation method
To find the PDF of Z=X/Y, we use the transformation method. We introduce an auxiliary variable, say W=Y, so that we have a one-to-one transformation from (X, Y) to (Z, W). From Z=X/Y, we can express X in terms of Z and W as X=ZW.
Step 3: Calculate the Jacobian of the transformation
The Jacobian of the transformation is needed to relate the joint PDF of (X, Y) to the joint PDF of (Z, W). We calculate the partial derivatives of x and y with respect to z and w. Since x=zw and y=w, the Jacobian determinant is w.
Step 4: Find the joint PDF of Z and W
The joint PDF of Z and W is obtained by substituting x=zw and y=w into fX,Y(x,y) and multiplying by the absolute value of the Jacobian. This gives us the joint density function in terms of z and w.
Step 5: Integrate out W to find the PDF of Z
To find the marginal PDF of Z, we integrate the joint PDF fZ,W(z,w) with respect to w over its entire range. Since w=Y and Y is a standard normal variable, w can take any real value.
Step 6: Evaluate the integral
The integral can be simplified by noting that the integrand is an even function of w, so we can integrate from 0 to ∞ and multiply by 2. We use a substitution u=w2(z2+1)/2, so du=w(z2+1)dw. Evaluating the definite integral yields the final PDF.