The random variable TTT represents the water temperature level (in ∘C^{\circ}\text{C}∘C) at a specific marine research site. The probability distribution for TTT is given in the following table:
| ttt | 2 | 5 | 8 | 10 |
|---|---|---|---|---|
| P(T=t)P(T=t)P(T=t) | 0.2 | 0.3 | 0.3 | 0.2 |
Show that E(T)=6.3E(T) = 6.3E(T)=6.3.
Find Var(T)Var(T)Var(T).
The random variable SSS represents the salinity index at the same site, recorded as a discrete index. The probability distribution for SSS is given in the following table:
| sss | 3 | 6 | 8 | kkk |
|---|---|---|---|---|
| P(S=s)P(S=s)P(S=s) | 0.25 | 0.25 | 0.25 | 0.25 |
Name the probability distribution of SSS.
Given that E(S)=E(T)E(S) = E(T)E(S)=E(T), find the value of kkk.
A specific species of bioluminescent jellyfish is found at a depth DDD meters, where D∼N(μ,σ2)D \sim N(\mu, \sigma^2)D∼N(μ,σ2). In a simulation, two researchers, Alice and Bob, each generate values for the parameters of the distribution. They each obtain a value for TTT to use as μ\muμ and a value for SSS to use as σ\sigmaσ. For each researcher, the values of TTT and SSS are generated independently according to the given distributions. The researcher whose parameters yield the higher probability P(D>7.0)P(D > 7.0)P(D>7.0) wins.
Alice obtained t=8t = 8t=8 and s=3s = 3s=3. Bob obtained s=6s = 6s=6. Find the probability that Bob wins.
Determine the largest value of P(D>7.0)P(D > 7.0)P(D>7.0) achievable in this simulation.
Find the probability of a researcher achieving this maximum value.