Two random variables X and W are distributed as follows X~N(6,6) and W~N(12,9).

Question

a. Find the distribution for the following linear transformations. i) X¡+W¡ ii) X¡-W¡ b) Find P(12.87< or equal to X¡ + W¡< or equal to 18.32)

Answer 1

(a)

(i) We can combine the means and variances using simple arithmetic: combined means=6+12=18, combined variances=6+9=15. So combining the datasets (we end up with a dataset, the size of which is the product of the sizes of the datasets) we can work out Z values=(X+W-18)/√15. We need the standard deviation of the combined set so we take the square root of the combined variance.

(ii) Combined means 6-12=-6, combined variances=6+9=15, because variance is always positive.

(b)

For X+W=12.87, Z=(12.87-18)/√15=-1.32456 approx.

For X+W=18.32, Z=(18.32-18)/√15=0.08262 approx.

From normal distribution tables, the probability for Z=-1.32 is 1-p(1.32)=1-0.9066=0.0934 and 0.5319 for Z=0.08. The probability of being within these limits is 0.5319-0.0934=0.4385 or 43.85%.