
Suppose that the random variable X** assumes three values 0, 1 and 2 with probabilities 1/3​, 1/6​ and 1/2​ respectively. Calculate the value of F(1)

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.

Learn how to calculate the cumulative distribution function (CDF) F(1) for a random variable X that takes values 0, 1, and 2 with probabilities 1/3, 1/6, and 1/2 respectively. This problem involves understanding probability theory and discrete random variables.

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