Let N_i ~ N(mu_i, sigma^2_i), i = 1, 2 be two independent normal random variable

Question

Let N_i ~ N(mu_i, sigma^2_i), i = 1, 2 be two independent normal random variables, with mu_1 > mu_2 and sigma_2 notequalto sigma_2. Define D N_1 - N_2. i. Show that D is a normal random variable, and calculate its mean and variance. ii. Let each N_i represent the output from one of two alternatives that we are comparing via simulation to select the system with the higher mean output. We do this by sampling from the difference D to estimate z = E[D], the mean difference and select system 1 if z > 0 and system 2 otherwise. How many replications would we need in order to obtain a 95% confidence guarantee that the difference is significant up to at least a given value Delta > 0?

Explanation / Answer

If N1 = (u1, 12), N2 =  (u2, 22 )

i ) N1 - N2

Usually, the sum or difference of two normal distribution is also normal distribution, i.e. when means are added are subtracted the resultant mean is the algebraic sum of two means, however, resultant standard deviation is root mean square of two standard deviations irrespective of the addition or subtraction

Hence resultant mean N = u1-u2

resultant standard devaition (s) = ( 12 + 22 )

ii) For difference of means to be greater than 95% confidence interval

at 95% CI, Z = 1.96 (Z table)

X = (( u1-u2 ) - 0 / ( / n))

1.96 = u1-u2 * n / ( 12 + 22 )

n = 3.8416* (u1-u2)^2 /  ( 12 + 22 )

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