The problem asks to construct the cumulative distribution function (CDF), $F(x)$, for a discrete random variable $X$. Also, it mentions constructing a graph. The given probability distribution is defined as: $P(X = 2) = 0.3$ $P(X = 4) = 0.1$ $P(X = 5) = 0.2$ $P(X = 6) = 0.4$ The CDF is given by $F(x) = P(X \le x)$.

1. Problem Description

The problem asks to construct the cumulative distribution function (CDF), $F(x)$, for a discrete random variable $X$. Also, it mentions constructing a graph. The given probability distribution is defined as:

$P(X = 2) = 0.3$

$P(X = 4) = 0.1$

$P(X = 5) = 0.2$

$P(X = 6) = 0.4$

The CDF is given by $F(x) = P(X \le x)$.

2. Solution Steps

First, we calculate the CDF for each value of $x$:

For $x < 2$, $F(x) = P(X \le x) = 0$

For $2 \le x < 4$, $F(x) = P(X \le x) = P(X = 2) = 0.3$

For $4 \le x < 5$, $F(x) = P(X \le x) = P(X = 2) + P(X = 4) = 0.3 + 0.1 = 0.4$

For $5 \le x < 6$, $F(x) = P(X \le x) = P(X = 2) + P(X = 4) + P(X = 5) = 0.3 + 0.1 + 0.2 = 0.6$

For $x \ge 6$, $F(x) = P(X \le x) = P(X = 2) + P(X = 4) + P(X = 5) + P(X = 6) = 0.3 + 0.1 + 0.2 + 0.4 = 1.0$

Therefore, the CDF $F(x)$ is a step function:

$F(x) = \begin{cases}

0, & x < 2 \\

0.3, & 2 \le x < 4 \\

0.4, & 4 \le x < 5 \\

0.6, & 5 \le x < 6 \\

1.0, & x \ge 6

\end{cases}$

3. Final Answer

The cumulative distribution function is:

$F(x) = \begin{cases}

0, & x < 2 \\

0.3, & 2 \le x < 4 \\

0.4, & 4 \le x < 5 \\

0.6, & 5 \le x < 6 \\

1.0, & x \ge 6

\end{cases}$