We use the notation X ~N(μ, σ2) to indicate that the density function for the continuous random variable X, fx(x), has the form
(a) If X ~N(μ, σ2) show that..... (Hint: you will need to know how to find the density function for X ? μ from the density function for X).
(b) If ...., and X1 and X2 are independent, prove that ..... (Hint: the density function for X1 + X2, fxi+x2(x), is given by the convolution formula
in the integrand, complete the square in the y-terms in the exponent, and then use an appropriate substitution and the fact that
to complete the proof.)
(c) If X1 N(p1,a) and X2 N(t2,c4), and X1 and X2 are independent, use parts (a) and (b) above to show that (X1 + X2) ~N(μ1 + μ2, ...).
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a) Since , the pdf is
For , we can compute its pdf as follows.
So, its pdf is
which implies that
b) We can ...
Stochastic Differential Equations, Density Functions and Random Variables are investigated. The solution is detailed and well presented.