The image shows a probability distribution function where $p(x)$ for different values of $x$ is expressed in terms of a constant $k$. To find the value of $k$, we will use the fact that the sum of all probabilities must be equal to 1.

Steps:

1. Set up the equation for normalization:

   $k + 3k + 5k + 7k + 9k + 11k + 13k = 1$

   Simplify this to find the value of $k$:

   $48k = 1 \quad \Rightarrow \quad k = \frac{1}{48}$

2. Calculate the variance $\text{Var}(X)$:

   The variance is calculated using the formula:

   [ \text{Var}(X) = E[X^2] - (E[X])^2 ]

   Where:

   • $E[X] = \sum x \cdot p(x)$
   • $E[X^2] = \sum x^2 \cdot p(x)$

   After finding $k$, we can substitute the values into these formulas to compute the mean $E[X]$ and then use it to find $E[X^2]$.

Let me know if you would like me to continue with the detailed calculations for the variance or if you'd like additional steps clarified.