An oceanographic research station utilizes two distinct models of deep-sea power probes, Alpha and Beta, for underwater data collection. The battery life, XXX hours, of a Model Alpha probe is normally distributed such that X∼N(120,25)X \sim N(120, 25)X∼N(120,25). The battery life, YYY hours, of a Model Beta probe is normally distributed such that Y∼N(128,16)Y \sim N(128, 16)Y∼N(128,16). The battery lives of the probes are independent.
Determine the probability that a Model Alpha probe, selected at random, outlasts a randomly selected Model Beta probe.
An engineer deploys 3 Model Alpha probes and 1 Model Beta probe.
Calculate the probability that the sum of the battery lives of the 3 Model Alpha probes is greater than 2.5 times the battery life of the Model Beta probe.
A composite metric SSS is defined as S=k1X+k2YS = k_1 X + k_2 YS=k1X+k2Y, where k1k_1k1 and k2k_2k2 are real constants. The engineer requires the mean of SSS to be exactly 3000 while ensuring the variance of SSS is minimized.
Determine the specific values of k1k_1k1 and k2k_2k2 that satisfy these requirements.
