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Two standard normal variables Z1 and Z2 are independent. Find the probability density function of the...

Two standard normal variables Z1 and Z2 are independent. Find the probability density function of the normal random variable X=3Z1 - Z2
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Answer #1

given Z1 and Z2 are independent standard normal random variables
so the linear combination of Z1 and Z2 is again follows normal distribution as follows

as = 0 and =0 and =1 and =1 ( where , are means and standard deviation of Zi respectively)

X=3Z1-Z2

has normal distribution with μ=3(0)-1(0)= 0

variance= =9*1+(-1)^2*1= 10

so the probability density function is given by :

where =0 ,    =10

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