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. 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.
