Suppose that either of two instruments might be used for making a certain measurement. Instrument 1 yields a measurement whose p.d.f. is
f1(x)=2x, 0 <x<1
Instrument 2 yields a measurement whose p.d.f. is
f2(x)=3x^2, 0 <x<1
Suppose that one of the two instruments is chosen at random and a measurement X is made with it.
(a)
Determine the marginal p.d.f. of X.
(b)
If X = 1/4 what is the probability that instrument 1 was used?
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To determine the marginal probability density function (p.d.f.) of X, we need to find the probability distribution of X when either Instrument 1 or Instrument 2 is used.
(a)
To find the marginal p.d.f. of X, we need to consider the probability of using each instrument and the respective p.d.f. associated with that instrument.
Let's assume the probability of using Instrument 1 is p1, and the probability of using Instrument 2 is p2. Since only one of the two instruments can be chosen, we have p1 + p2 = 1.
For a given value x in the range (0, 1), we can express the overall probability of obtaining that value as:
P(X = x) = P(X = x|using Instrument 1) * P(using Instrument 1) +
P(X = x|using Instrument 2) * P(using Instrument 2)
For Instrument 1, P(X = x|using Instrument 1) = f1(x) = 2x, and P(using Instrument 1) = p1.
For Instrument 2, P(X = x|using Instrument 2) = f2(x) = 3x^2, and P(using Instrument 2) = p2.
So, the marginal p.d.f. of X can be expressed as:
fX(x) = P(X = x) = 2x * p1 + 3x^2 * p2, for 0 < x < 1
(b)
To find the probability that Instrument 1 was used given that X = 1/4, we can use Bayes' theorem. Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, A represents the event of Instrument 1 being used, and B represents the event of X = 1/4.
P(A|B) is the probability of Instrument 1 being used given that X = 1/4, P(B|A) is the probability of X = 1/4 given that Instrument 1 was used, P(A) is the probability of Instrument 1 being chosen (p1), and P(B) is the overall probability of X = 1/4.
We can use the marginal p.d.f. of X to find P(B):
P(B) = fX(1/4) = 2 * (1/4) * p1 + 3 * (1/4)^2 * p2
Now, substituting the given value of x into the marginal p.d.f. will give us P(B).
Once you have calculated P(B), you can substitute the values into Bayes' theorem to calculate P(A|B), which represents the probability that Instrument 1 was used given X = 1/4.