Statistics

Step 1: Probability of getting an even number.
The even numbers on a die are . So, the probability of getting an even number is Step 2: Use the binomial distribution formula.
The standard deviation for a binomial distribution is given by where is the number of trials, is the probability of success, and .
Step 3: Calculate the standard deviation.
Here, , , and . Thus, Step 4: Conclusion.
The standard deviation is 5.

To find the mean and variance of natural numbers:

Step 1: Mean
The sum of the first natural numbers is given by the formula: To calculate the mean, we divide the sum by . Therefore, the mean of the first natural numbers is: Thus, the mean is .

Step 2: Variance
The variance is a measure of how much the numbers deviate from the mean. It can be calculated using the formula: However, an alternative formula for the variance of the first natural numbers is given by: Simplifying this expression gives:

Final Answer:
Hence, the correct option for the variance is:

Step 1: Find the Mean of the First 2n Natural Numbers
The mean is the sum of the numbers divided by the total count of numbers:

Step 2: Simplify the Mean Using the Arithmetic Series Formula
Using the sum formula for an arithmetic series, we get: Simplifying the mean, we get:

Step 3: Define the Variance
Variance is defined as the average of squared differences from the mean:

Step 4: Simplify the Variance Expression
Expanding and simplifying the expression, we obtain:

Step 5: Further Simplify the Variance
This simplifies to:

Step 6: Final Result
Therefore, the variance of the first 2n natural numbers is:

Step 1: Given Information
We are given the following: Number of observations
We need to find the value of .

Step 2: Find the Mean :

Step 3: Simplify the Given Sum
The given sum looks like the formula for variance, but with 'x' instead of the mean: Expanding the summation: Substitute values:

Step 4: Use Variance Formula
We know the formula for variance : We also know: Thus, we have: Substituting : Now substitute into the equation: Simplifying: This equation is always true, meaning can be any real number.

Step 5: Conclusion
To find a simpler solution, assume . Then: Since this satisfies the given condition, the most likely value of is 1.

Final Answer: The value of is 1.

1. Simplify the Integrand:
We are given the equation: where is the mean and is the variance.

2. Apply the Relationship Between Variance and Mean:
The relationship is given. Substituting this into the sum equation:

3. Solve for :
Multiply through by , factor out , and solve: Substituting :

4. Calculate :
Substitute into :

Final Answer:
Therefore, the variance is:

Step 1: Determine the total number of possible outcomes. When selecting and from the set with replacement, the possible pairs are: Thus, the total number of outcomes is: Step 2: Identify the favorable outcomes where . List the pairs satisfying : The number of favorable outcomes is: Step 3: Calculate the probability. The probability of the event is given by: Final Answer: The probability is: