The random variable WWW represents the initial nutrient concentration, in mg/L, in a series of botanical samples. The probability distribution for WWW is given in the following table:
| www | 1 | 3 | 6 | 8 |
|---|---|---|---|---|
| P(W=w)P(W=w)P(W=w) | 0.45 | 0.2 | 0.2 | 0.15 |
Show that E(W)=3.45E(W) = 3.45E(W)=3.45.
Find Var(W)Var(W)Var(W).
The random variable SSS represents the soil porosity index of the sample's medium. The probability distribution for SSS is given in the following table, where kkk is a constant:
| sss | 2 | 4 | 5 | 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(W)E(S) = E(W)E(S)=E(W), find the value of kkk.
The growth of a seedling, GGG mm, is modelled by the normal distribution G∼N(μ,σ2)G \sim N(\mu, \sigma^2)G∼N(μ,σ2). Researchers Alice and Bob each select a nutrient concentration for μ\muμ and a porosity index for σ\sigmaσ by sampling from the distributions of WWW and SSS respectively. A sample is considered 'successful' if its growth exceeds 5 mm. The researcher whose parameters result in a higher probability of success, P(G>5)P(G > 5)P(G>5), wins.
Alice obtained w=6w = 6w=6 and s=2s = 2s=2. Bob obtained s=5s = 5s=5. Determine the probability that Bob wins.
Find the largest probability of success, P(G>5)P(G > 5)P(G>5), achievable in this experiment.
Find the probability of a researcher achieving this maximum probability of success.
