Math Problem Statement
4.2. Suppose that takes on one of the values 0, 1, and 2. If for some constant c P(x=i) = cp(x=i-1), i=1,2 , find . E[x]
Solution
To solve this, let's follow the steps systematically:
Step 1: Given Conditions
- takes values .
- The probability for .
- The total probability must sum to 1: .
Let:
- ,
- ,
- .
Step 2: Total Probability Equation
From the total probability: Factor out : Solve for :
Step 3: Probabilities
- ,
- ,
- .
Step 4: Expected Value
The expected value is given by: Substitute the probabilities: Simplify:
Final Answer
Let me know if you want a detailed example or graph to visualize this!
Here are 5 questions for further exploration:
- What happens to as approaches 0 or infinity?
- Can this method extend to higher values of ?
- What are the properties of when ?
- How does the variance of relate to ?
- Can we generalize this distribution form to continuous random variables?
Tip: Always verify that probabilities sum to 1, especially when dealing with parameterized distributions!
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Math Problem Analysis
Mathematical Concepts
Probability
Expected Value
Discrete Random Variables
Formulas
P(X=i) = c * P(X=i-1)
E[X] = Σ(i * P(X=i))
Theorems
Total Probability Theorem
Expected Value Formula
Suitable Grade Level
Grades 11-12
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