Probability And Statistics

Step 1: Understanding the Concept:

This question tests the knowledge of finding the variance of a discrete random variable given its probability distribution. First, we must find the value of the unknown 'k' using the property that the sum of all probabilities in a distribution is 1. Then, we calculate the mean (Expected Value, ) and the expectation of the square of the variable ( ). Finally, we use the variance formula.

Step 2: Key Formula or Approach:

1. Sum of probabilities: .
2. Mean (Expected Value): .
3. Expectation of : .
4. Variance: .

Step 3: Detailed Explanation:

Find k:
The sum of all probabilities must be 1. Now we can complete the probability distribution table:

Calculate Mean :

Calculate :

Calculate Variance :

Convert to Fraction:
Now, let's check the options.
Option (A): .
Let's calculate this value: .
This matches our calculated variance.

Step 4: Final Answer:

The variance of the random variable is 1.2224, which is equal to .

Step 1: Understanding the Concept:

This question tests the knowledge of finding the variance of a discrete random variable given its probability distribution. First, we must find the value of the unknown 'k' using the property that the sum of all probabilities in a distribution is 1. Then, we calculate the mean (Expected Value, ) and the expectation of the square of the variable ( ). Finally, we use the variance formula.

Step 2: Key Formula or Approach:

1. Sum of probabilities: .
2. Mean (Expected Value): .
3. Expectation of : .
4. Variance: .

Step 3: Detailed Explanation:

Find k:
The sum of all probabilities must be 1. Now we can complete the probability distribution table:

Calculate Mean :

Calculate :

Calculate Variance :

Convert to Fraction:
Now, let's check the options.
Option (A): .
Let's calculate this value: .
This matches our calculated variance.

Step 4: Final Answer:

The variance of the random variable is 1.2224, which is equal to .