Math Problem Statement

Let's solve for 𝑓 𝑋 ∣ π‘Œ

0 , 𝑍

1 ( 0 ) f X∣Y=0,Z=1 ​ (0) step by step using the information provided.

Step 1: Bayes' Theorem for Conditional Probability We want to find:

𝑓 𝑋 ∣ π‘Œ

0 , 𝑍

1 ( 0 )

𝑓 𝑋 π‘Œ 𝑍 ( 0 , 0 , 1 ) 𝑓 π‘Œ 𝑍 ( 0 , 1 ) . f X∣Y=0,Z=1 ​ (0)= f YZ ​ (0,1) f XYZ ​ (0,0,1) ​ . Step 2: Marginalization to Find 𝑓 π‘Œ 𝑍 ( 0 , 1 ) f YZ ​ (0,1) To compute 𝑓 π‘Œ 𝑍 ( 0 , 1 ) f YZ ​ (0,1), we marginalize over 𝑋 X:

𝑓 π‘Œ 𝑍 ( 0 , 1 )

𝑓 𝑋 π‘Œ 𝑍 ( 0 , 0 , 1 ) + 𝑓 𝑋 π‘Œ 𝑍 ( 1 , 0 , 1 ) . f YZ ​ (0,1)=f XYZ ​ (0,0,1)+f XYZ ​ (1,0,1). We already know:

𝑓 𝑋 π‘Œ 𝑍 ( 0 , 0 , 1 )

1 4 f XYZ ​ (0,0,1)= 4 1 ​ . However, we don't have 𝑓 𝑋 π‘Œ 𝑍 ( 1 , 0 , 1 ) f XYZ ​ (1,0,1) directly, but let's compute it based on the Bernoulli distributions.

Step 3: Use of Bernoulli Distributions 𝑋 ∼ Bernoulli ( 1 / 3 ) X∼Bernoulli(1/3), so 𝑃 ( 𝑋

1 )

1 3 P(X=1)= 3 1 ​ and 𝑃 ( 𝑋

0 )

2 3 P(X=0)= 3 2 ​ . π‘Œ ∼ Bernoulli ( 1 / 2 ) Y∼Bernoulli(1/2), so 𝑃 ( π‘Œ

0 )

1 2 P(Y=0)= 2 1 ​ . 𝑍 ∼ Bernoulli ( 2 / 3 ) Z∼Bernoulli(2/3), so 𝑃 ( 𝑍

1 )

2 3 P(Z=1)= 3 2 ​ . Step 4: Compute 𝑓 π‘Œ 𝑍 ( 0 , 1 ) f YZ ​ (0,1) Now, from the marginalization formula:

𝑓 π‘Œ 𝑍 ( 0 , 1 )

𝑓 𝑋 π‘Œ 𝑍 ( 0 , 0 , 1 ) + 𝑓 𝑋 π‘Œ 𝑍 ( 1 , 0 , 1 ) . f YZ ​ (0,1)=f XYZ ​ (0,0,1)+f XYZ ​ (1,0,1). Assume the conditional probabilities follow the Bernoulli distributions, which means:

𝑓 𝑋 π‘Œ 𝑍 ( 1 , 0 , 1 )

𝑃 ( 𝑋

1 ) β‹… 𝑃 ( π‘Œ

0 ) β‹… 𝑃 ( 𝑍

1 )

1 3 β‹… 1 2 β‹… 2 3

1 9 . f XYZ ​ (1,0,1)=P(X=1)β‹…P(Y=0)β‹…P(Z=1)= 3 1 ​ β‹… 2 1 ​ β‹… 3 2 ​

9 1 ​ . Thus, we now have:

𝑓 π‘Œ 𝑍 ( 0 , 1 )

1 4 + 1 9

9 + 4 36

13 36 . f YZ ​ (0,1)= 4 1 ​ + 9 1 ​

36 9+4 ​

36 13 ​ . Step 5: Calculate 𝑓 𝑋 ∣ π‘Œ

0 , 𝑍

1 ( 0 ) f X∣Y=0,Z=1 ​ (0) Now, we can substitute this into the formula for 𝑓 𝑋 ∣ π‘Œ

0 , 𝑍

1 ( 0 ) f X∣Y=0,Z=1 ​ (0):

𝑓 𝑋 ∣ π‘Œ

0 , 𝑍

1 ( 0 )

1 4 13 36

1 4 β‹… 36 13

9 13 . f X∣Y=0,Z=1 ​ (0)= 36 13 ​

4 1 ​

​

4 1 ​ β‹… 13 36 ​

13 9 ​ . Final Answer: The value of 𝑓 𝑋 ∣ π‘Œ

0 , 𝑍

1 ( 0 ) f X∣Y=0,Z=1 ​ (0) is approximately:

0.69 . 0.69 ​ .answer is 0.75

Solution

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Math Problem Analysis

Mathematical Concepts

Conditional Probability
Marginalization
Bayes' Theorem
Bernoulli Distributions

Formulas

f_{X \mid Y = 0, Z = 1}(0) = f_{XYZ}(0, 0, 1) / f_{YZ}(0, 1)
f_{YZ}(0, 1) = f_{XYZ}(0, 0, 1) + f_{XYZ}(1, 0, 1)
f_{XYZ}(x, y, z) = P(X = x) * P(Y = y) * P(Z = z)

Theorems

Bayes' Theorem
Law of Total Probability

Suitable Grade Level

University Level - Probability and Statistics

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