# Given Information - The problem statement or given data is **not provided** in your message. - Please supply the actual question or values for a specific solution. # What Have To Find - The required unknown(s) or result(s) to be determined are **not specified** due to missing problem data. # Definition or Concept Used - The relevant formula, theorem, or concept will depend on the specific problem you provide. - Common examples: - For motion: \( s = ut + \frac{1}{2}at^2 \) - For Ohm’s Law: \( V = IR \) - For area of a triangle: \( A = \frac{1}{2}bh \) # Step-by-Step Solution - A detailed, logical sequence for solving the problem will be shown here **after you provide the question and numbers**. # Summary (Final Answer) - The concise result will be written here **once the problem is provided and solved**. --- **Note:** Please provide the full question or numerical data so a structured solution can be written following your requested format.

# Given Information - A discrete random variable \( x \) with probability mass function: \[ P(x = k) = c k (k + 1) \] for \( k = 1, 2, 3, \ldots \) # What Have To Find 1. The value of \( c \). 2. The expected value \( E(x) \). # Definition or Concept Used 1. **Normalization of Probability**: The sum of all probabilities must equal 1: \[ \sum_{k=1}^{\infty} P(x = k) = 1 \] 2. **Expected Value**: The expected value for a discrete random variable is given by: \[ E(x) = \sum_{k=1}^{\infty} k \cdot P(x = k) \] # Step-by-Step Solution To find \( c \), we normalize the probability mass function: - Start with the sum of probabilities: \[ \sum_{k=1}^{\infty} P(x = k) = \sum_{k=1}^{\infty} c k (k + 1) = 1 \] - Simplify the expression: \[ \sum_{k=1}^{\infty} k (k + 1) = \sum_{k=1}^{\infty} (k^2 + k) = \sum_{k=1}^{\infty} k^2 + \sum_{k=1}^{\infty} k \] - Known formulas for infinite sums: - \( \sum_{k=1}^{\infty} k = \frac{n(n+1)}{2} \) diverges. - \( \sum_{k=1}^{\infty} k^2 = \frac{n(n+1)(2n+1)}{6} \) diverges. Thus: \[ \sum_{k=1}^{\infty} k^2 + \sum_{k=1}^{\infty} k = \infty \] This indicates that \( c \) must be adjusted to ensure convergence. To find \( c \), we realize: \[ c \sum_{k=1}^{\infty} k (k + 1) = c \cdot \infty = 1 \implies c = 0 \] This means, realistically, we need to limit \( k \) to a finite range or redefine \( P(x = k) \). Next, to find \( E(x) \): \[ E(x) = \sum_{k=1}^{\infty} k \cdot P(x = k) = c \sum_{k=1}^{\infty} k^2 (k + 1) \] Again, we find that without a finite domain, this will diverge. # Summary (Final Answer) - The value of \( c \) cannot be determined under infinite conditions; it suggests a need for finite \( k \). - The expected value \( E(x) \) diverges under the provided function.