![Design an image of a clean, educational setting. Highlight a whiteboard that is blank, save for some discrete mathematics and probability theory symbols aesthetically arranged. The symbols should include those typically used to represent random variables, value ranges, and conditional probability such as 'X', 'K', '[0,1]', and 'fX|K'. However, ensure there is no specific readable text or clear equations on the board. Additionally, depict placeholders for the values 1, 2, 3, to indicate different conditions, but without connecting them explicitly. The whiteboard is bathed in soft, inviting light and the atmosphere of the empty room suggests a quiet place for focused thinking.](/images/questions/1836376/1024x1024.png)
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.