To calculate the value of the cumulative distribution function (CDF) $F(x)$ at $x = 1$ for a discrete random variable $X$, we need to find:

$F(1) = P(X \leq 1)$

Given the random variable $X$ assumes three values: $0, 1,$ and $2$ with the following probabilities:

• $P(X = 0) = \frac{1}{3}$
• $P(X = 1) = \frac{1}{6}$
• $P(X = 2) = \frac{1}{2}$

Step-by-Step Solution

The cumulative distribution function $F(x)$ at $x = 1$ is defined as:

$F(1) = P(X \leq 1) = P(X = 0) + P(X = 1)$

Substitute the given probabilities:

$F(1) = \frac{1}{3} + \frac{1}{6}$

Now, let's find a common denominator to add these fractions:

$F(1) = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$

Final Answer

The value of $F(1)$ is $\frac{1}{2}$.

Would you like further details on this solution? Here are some related questions to explore:

1. What is the value of $F(2)$ for this random variable?
2. How would you calculate $F(x)$ for any $x$ not equal to 0, 1, or 2?
3. What is the probability mass function (PMF) of $X$?
4. How would the CDF change if the probabilities were different?
5. What does the CDF graph of this random variable look like?

Tip: The CDF $F(x)$ for a discrete random variable is a step function that only increases at the points where the variable takes specific values.