Math Problem Statement

For the discrete probability distribution table provided with x = {0, 1, 2, 3, 4, 5, 6} and P(x) = {0, 2k, 2k, 3k, k^2, 2k^2, 7k^2 + fk}, find k, f, the mean, and variance.

Solution

Step-by-step solution to the problem:

Given:

  1. The table of probabilities P(x)P(x) for a discrete random variable xx:

    • x=0,1,2,3,4,5,6x = 0, 1, 2, 3, 4, 5, 6
    • Corresponding probabilities P(x)=0,2k,2k,3k,k2,2k2,7k2+fkP(x) = 0, 2k, 2k, 3k, k^2, 2k^2, 7k^2 + fk
  2. Tasks:

    • Find kk,
    • Find ff,
    • Calculate the Mean,
    • Calculate the Variance.

Step 1: Probability Sum Property

For a valid probability distribution, the total probabilities must sum to 1: P(x)=1\sum P(x) = 1 Substitute the given probabilities: 0+2k+2k+3k+k2+2k2+(7k2+fk)=10 + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + fk) = 1 Simplify: 7k + 10k^2 + fk = 1 \tag{1}


Step 2: Solve for kk and ff

From the problem, there are two unknowns (kk and ff), and another condition may need to be established to uniquely solve for them. Is there an additional relationship between kk and ff?
Let me know if I should clarify this further or if there are additional constraints for ff. I'll pause to confirm this before proceeding to the next steps.


Do you want me to continue by solving for kk, ff, Mean, and Variance explicitly? Let me know!

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Math Problem Analysis

Mathematical Concepts

Probability Distribution
Algebra
Statistics

Formulas

Sum of probabilities: ΣP(x) = 1
Mean of a discrete distribution: μ = Σ[x * P(x)]
Variance of a discrete distribution: σ^2 = Σ[x^2 * P(x)] - μ^2

Theorems

Probability axioms

Suitable Grade Level

Grades 10-12