Question

Which of the following probability distribution functions (PDFs) has the mean greater than the median? \includegraphics[width=0.75\linewidth]{image3.png}

• A. Function 1
• B. Function 2
• C. Function 3
• D. Function 4

Hint:

For skewed distributions, remember:
Right-skewed: mean $>$ median $>$ mode.
Left-skewed: mean $<$ median $<$ mode.
This is a quick way to compare mean and median without calculation.

Answer 1

The Correct Option is B

Step 1: Recall the relationship between mean and median in skewed distributions.
If a distribution is symmetrical, the mean = median = mode.
If a distribution is positively skewed (right-skewed), the mean is greater than the median.
If a distribution is negatively skewed (left-skewed), the mean is less than the median.

Step 2: Examine each function.
Function 1: Symmetrical bell-shaped curve $\Rightarrow$ mean = median.
Function 2: Right-skewed (long tail to the right) $\Rightarrow$ mean $>$ median.
Function 3: Left-skewed (long tail to the left) $\Rightarrow$ mean $<$ median.
Function 4: Bimodal and roughly symmetric $\Rightarrow$ mean $\approx$ median.

Step 3: Conclusion.
Only Function 2 shows positive skewness, so its mean is greater than its median.
\[ \boxed{\text{Function 2}} \]