Let K be a discrete random variable that can take the values 1 , 2 , and 3 , all with equal probability. Suppose that X takes values in [0,1] and that for x in that interval we have
fX|K(x|k)=⎧⎩⎨1,2x,3x2,if k=1,if k=2,if k=3.
Find the probability that K=1 , given that X=1/2 .
4/11
Y is not a valid variable in the answer...
Well, let's think about this. We want to find the probability that K=1, given that X=1/2.
To do this, we can use Bayes' Theorem, which states that the probability of an event A given that event B has occurred is equal to the probability of event B given that event A has occurred times the probability of event A, divided by the probability of event B.
In this case, event A is K=1, and event B is X=1/2. So, we want to find the probability of K=1 given that X=1/2.
Using Bayes' Theorem, we have:
P(K=1|X=1/2) = P(X=1/2|K=1) * P(K=1) / P(X=1/2)
Now, we are given that for X=1/2, fX|K(x|k) = 2x if k=1. So, in this case, P(X=1/2|K=1) = 2(1/2) = 1.
We are also given that K can take the values 1, 2, and 3 with equal probability. So, P(K=1) = 1/3.
Now, let's find P(X=1/2). Since X takes values in the interval [0,1], we have to consider all the possible values of K for X=1/2.
If K=1, then fX|K(x|k) = 2x, so fX|K(1/2|1) = 2(1/2) = 1.
If K=2, then fX|K(x|k) = x, so fX|K(1/2|2) = 1/2.
If K=3, then fX|K(x|k) = 3x^2, so fX|K(1/2|3) = 3(1/2)^2 = 3/4.
So, to find P(X=1/2), we have to consider all the possible values of K:
P(X=1/2) = P(X=1/2|K=1) * P(K=1) + P(X=1/2|K=2) * P(K=2) + P(X=1/2|K=3) * P(K=3)
= 1 * 1/3 + 1/2 * 1/3 + 3/4 * 1/3
= 1/3 + 1/6 + 1/4
= 19/36
Now, substituting back into Bayes' Theorem, we have:
P(K=1|X=1/2) = 1 * (1/3) / (19/36)
Dividing 1/3 by 19/36, we get:
P(K=1|X=1/2) = 12/19
So, the probability that K=1, given that X=1/2, is 12/19.
To find the probability that K=1 given that X=1/2, we can use Bayes' Theorem. Bayes' Theorem is a formula used in probability theory to calculate the probability of an event occurring, based on certain prior knowledge or conditions.
Bayes' Theorem states that:
P(A|B) = P(B|A) * P(A) / P(B)
Where:
P(A|B) is the probability of event A occurring, given that event B has already occurred.
P(B|A) is the probability of event B occurring, given that event A has already occurred.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occurring.
In this case, we want to find the probability that K=1, given that X=1/2. So, we can rewrite the problem as finding P(K=1|X=1/2).
Using Bayes' Theorem, we have:
P(K=1|X=1/2) = P(X=1/2|K=1) * P(K=1) / P(X=1/2)
Now let's break down the parts of the equation:
P(X=1/2|K=1) is the probability of X=1/2, given that K=1. From the given information, we know that fX|K(x|k) = 1 for k=1. So, P(X=1/2|K=1) = 1.
P(K=1) is the probability of K=1. Since K is a discrete random variable that can take the values 1, 2, and 3 with equal probability, P(K=1) = 1/3.
P(X=1/2) is the probability of X=1/2. To find this probability, we need to calculate the probability density function (pdf) of X and integrate over the interval [0,1]. Using the given information, we can see that for x in the interval [0,1], fX(x) = fX|K(x|1) * P(K=1) + fX|K(x|2) * P(K=2) + fX|K(x|3) * P(K=3). Given that fX|K(x|k) = 1 for k=1, P(K=1) = 1/3, and integrating over the interval [0,1], we have:
P(X=1/2) = 1/3 * 1 + 0 + 0 = 1/3.
Substituting these values into the equation, we get:
P(K=1|X=1/2) = 1 * (1/3) / (1/3)
Simplifying the expression, we get:
P(K=1|X=1/2) = 1.
Therefore, the probability that K=1, given that X=1/2, is 1.