Let X be a random variable following the standard normal dis

Let X be a random variable following the standard normal distribution. Let Z = X^2 + 2X. Find the PDF of Y, its mean and its variance

Solution

Let\'s write P(X)=N(X) ...#1

(Where N(X) is the normal distribution function over X)

Z=z => X(X+2) = z => (X+1)²-1 = z => X=-1-(z+1)^0.5 OR X=-1+(z+1)^0.5  

Let\'s assume X_min=-1-(z+1)^0.5

  X_max=-1+(z+1)^0.5

We can use a simple tranformation formula

PDF(Z) = PDF(X at where Z=z) * (|dx/dz|)

|dz/dx|=2(x+2)

=>|dx/dz|=0.5(x+2)^-1

Now PDF(Z) = PDF(X at X=X_min)*(0.5(X_min+2)^-1) + PDF(X at X=X_max)*(0.5(X_max+2)^-1)

= N(X_min)*(0.5(X_min+2)^-1) + N(X_max)*(0.5(X_max+2)^-1)

=0.5*[ N(X_min) * (X_min+2)^-1 + N(X_max) * (X_max+2)^-1 ] ......ANSWER

Where N(X)=Normal Distribution Function Over X

X_min=-1-(z+1)^0.5

X_max=-1+(z+1)^0.5