A random variable X has the following probability function for 
various values of 𝑥
𝑥 0 1 2 3 4 5 6 7
𝑓(𝑥) 0, 𝑘, 2𝑘 2𝑘, 3𝑘, 𝑘^2, 2k^2, 7k^2 +k
(i)Find 𝑘 (ii) Evaluate 𝑃(𝑥) < 6 & 𝑃(3 < 𝑥 ≤ 6).

To solve the problem, we need to follow these steps:

### (i) Find the value of \( k \)

Given the probability function \( f(x) \) for the random variable \( X \) over different values of \( x \):

\[
\begin{aligned}
x & : 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
f(x) & : 0 & k & 2k & 2k & 3k & k^2 & 2k^2 & 7k^2 + k
\end{aligned}
\]

The sum of all probabilities must equal 1:

\[
f(0) + f

Explore how to find the value of k in a probability function and evaluate probabilities for a random variable X. This problem involves calculating probabilities using a given probability function and finding the constant k. Suitable for college-level mathematics.

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