Suppose X_1 is an observation for Bob, X_5 is an observation for Alice, X_7 is an observation for Charlie.
Using the following facts about Bob:
P( -2 < X_1 < 2 ) \approx 0.95 \quad \text {and} \quad P(-\sqrt[5]{2} \le X_1 \le \sqrt[5]{2}) \approx 0.75,
compute the probability
P( -2 < X_5 < 2, \; -2 < (X_7)^5 < 2)
of an event involving Alice and Charlie?
(Enter the probability P( -2 < X_5 < 2, \; -2 < (X_7)^5 < 2) or if the probability is not determined uniquely, then enter -1.)
(Enter a numerical answer accurate to at least 3 decimal places.)
P( -2 < X_5 < 2, \; -2 < (X_7)^5 < 2)=
Since the problem only provides information about Bob and not about Alice or Charlie, we cannot directly determine the probability of the event involving Alice and Charlie. Therefore, the probability is not determined uniquely, and the answer is -1.
To compute the probability of the event involving Alice and Charlie, we need to use the fact about Bob to estimate the probabilities for Alice and Charlie.
From the given fact about Bob:
P(-2 < X_1 < 2) ≈ 0.95
P(-∛2 ≤ X_1 ≤ ∛2) ≈ 0.75
We can assume that the observations for Alice, X_5, and Charlie, X_7, are independent and have the same distribution and properties as Bob's observation, X_1. Therefore, the probabilities for Alice and Charlie can be estimated as follows:
P(-2 < X_5 < 2) ≈ 0.95
P(-∛2 ≤ X_5 ≤ ∛2) ≈ 0.75
P(-2 < (X_7)^5 < 2) ≈ 0.95
P(-∛2 ≤ (X_7)^5 ≤ ∛2) ≈ 0.75
Now, to compute the probability of the event involving Alice and Charlie, we need to find the intersection of the two events:
P(-2 < X_5 < 2, -2 < (X_7)^5 < 2) = P(-2 < X_5 < 2) * P(-2 < (X_7)^5 < 2)
Substituting the estimated probabilities:
P(-2 < X_5 < 2, -2 < (X_7)^5 < 2) ≈ 0.95 * 0.95
Therefore, the probability of the event involving Alice and Charlie is approximately 0.9025.
Answer: P(-2 < X_5 < 2, -2 < (X_7)^5 < 2) ≈ 0.9025