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

  • XX takes values 0,1,20, 1, 2.
  • The probability P(X=i)=cP(X=i1)P(X = i) = c \cdot P(X = i-1) for i=1,2i = 1, 2.
  • The total probability must sum to 1: P(X=0)+P(X=1)+P(X=2)=1P(X = 0) + P(X = 1) + P(X = 2) = 1.

Let:

  • P(X=0)=p0P(X = 0) = p_0,
  • P(X=1)=p1=cp0P(X = 1) = p_1 = c \cdot p_0,
  • P(X=2)=p2=cp1=c2p0P(X = 2) = p_2 = c \cdot p_1 = c^2 \cdot p_0.

Step 2: Total Probability Equation

From the total probability: p0+cp0+c2p0=1.p_0 + c \cdot p_0 + c^2 \cdot p_0 = 1. Factor out p0p_0: p0(1+c+c2)=1.p_0 (1 + c + c^2) = 1. Solve for p0p_0: p0=11+c+c2.p_0 = \frac{1}{1 + c + c^2}.

Step 3: Probabilities

  • P(X=0)=p0=11+c+c2P(X = 0) = p_0 = \frac{1}{1 + c + c^2},
  • P(X=1)=p1=cp0=c1+c+c2P(X = 1) = p_1 = c \cdot p_0 = \frac{c}{1 + c + c^2},
  • P(X=2)=p2=c2p0=c21+c+c2P(X = 2) = p_2 = c^2 \cdot p_0 = \frac{c^2}{1 + c + c^2}.

Step 4: Expected Value E[X]E[X]

The expected value is given by: E[X]=i=02iP(X=i).E[X] = \sum_{i=0}^{2} i \cdot P(X = i). Substitute the probabilities: E[X]=0P(X=0)+1P(X=1)+2P(X=2).E[X] = 0 \cdot P(X = 0) + 1 \cdot P(X = 1) + 2 \cdot P(X = 2). E[X]=0+c1+c+c2+2c21+c+c2.E[X] = 0 + \frac{c}{1 + c + c^2} + 2 \cdot \frac{c^2}{1 + c + c^2}. Simplify: E[X]=c+2c21+c+c2.E[X] = \frac{c + 2c^2}{1 + c + c^2}.

Final Answer

E[X]=c+2c21+c+c2.E[X] = \frac{c + 2c^2}{1 + c + c^2}.


Let me know if you want a detailed example or graph to visualize this!

Here are 5 questions for further exploration:

  1. What happens to E[X]E[X] as cc approaches 0 or infinity?
  2. Can this method extend to higher values of XX?
  3. What are the properties of E[X]E[X] when c=1c = 1?
  4. How does the variance of XX relate to cc?
  5. 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