Mean And Variance Of Random Variables

Step 1: List the First 10 Prime Numbers

The first 10 prime numbers are:

Step 2: Find the Median

Since there are 10 numbers (even count), the median is the average of the 5th and 6th terms:

Final Answer:

Step 1: Understanding Ogive Intersection

In statistics, the intersection of the less than ogive and the more than ogive gives the median of the data.

Step 2: Identify the Median

From the given data, the two ogives intersect at . The x-coordinate of this point represents the median.

Final Answer:

To solve this problem, we will use the empirical relationship between mean, median, and mode typically observed in a moderately skewed distribution.
This relationship is given by: Mode = 3 Median - 2 Mean

Let's denote:

• M_o = Mode
• M_e = Median
• X̄ = Mean

According to the problem, the difference between mode and mean is k times the difference between median and mean:
(M_o - X̄) = k(M_e - X̄)
Substitute the empirical relationship into the equation:
(3M_e - 2X̄ - X̄) = k(M_e - X̄)
3M_e - 3X̄ = kM_e - kX̄
Rearranging terms gives us:
3M_e - kM_e = 3X̄ - kX̄
M_e(3 - k) = X̄(3 - k)
Since M_e and X̄ are not equal in a skewed distribution, we cannot cancel (3 - k).
Therefore, if 3 - k = 0, k = 3.
Thus, the value of k is 3.