The probability function of a random variable X is given by
`p(x)=(1)/(3)`, if x = -1, 0, 1
=0, otherwise.
Find the distribution function of X.

Answer

To find the distribution function of the random variable \( X \) with the given probability function, we will follow these steps:

### Step 1: Define the Probability Function
The probability function \( p(x) \) is defined as follows:
\[ p(x) = \begin{cases} \frac{1}{3} & \text{if } x = -1, 0, 1 \\ 0 & \text{otherwise} \end{cases} \]

### Step 2: Identify the Values of \( X \)
The random variable \( X \) can take the values:
- \( -1 \)
- \( 0 \)
- \( 1 \)

### Step 3: Calculate the Cumulative Distribution Function (CDF)
The cumulative distribution function \( F(x) \) is defined as:
\[ F(x) = P(X \leq x) \]
We will calculate \( F(x) \) for different intervals of \( x \).

1. **For \( x < -1 \)**:
\[ F(x) = 0 \quad (\text{since } P(X \leq x) = 0) \]

2. **For \( -1 \leq x < 0 \)**:
\[ F(x) = P(X \leq -1) = p(-1) = \frac{1}{3} \]

3. **For \( 0 \leq x < 1 \)**:
\[ F(x) = P(X \leq 0) = P(X = -1) + P(X = 0) = p(-1) + p(0) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \]

4. **For \( x \geq 1 \)**:
\[ F(x) = P(X \leq 1) = P(X = -1) + P(X = 0) + P(X = 1) = p(-1) + p(0) + p(1) = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1 \]

### Step 4: Summarize the CDF
Now we can summarize the cumulative distribution function \( F(x) \):
\[ F(x) = \begin{cases} 0 & \text{if } x < -1 \\ \frac{1}{3} & \text{if } -1 \leq x < 0 \\ \frac{2}{3} & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases} \]

### Final Answer
The distribution function \( F(x) \) of the random variable \( X \) is:
\[ F(x) = \begin{cases} 0 & \text{if } x < -1 \\ \frac{1}{3} & \text{if } -1 \leq x < 0 \\ \frac{2}{3} & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases} \]