To construct the relative frequency marginal distribution, we need to calculate the relative frequencies for each element by dividing the given frequencies by the grand total (which is the sum of all the values in the table).

Step 1: Find the grand total

The grand total is the sum of all the values in the table:

$20 + 25 + 30 + 30 + 15 + 50 = 170$

Step 2: Calculate the relative frequencies

Now, divide each individual value by the grand total (170), and round the results to three decimal places.

For $y_1$ row:

• $x_1$: $\frac{20}{170} = 0.118$
• $x_2$: $\frac{25}{170} = 0.147$
• $x_3$: $\frac{30}{170} = 0.176$

For $y_2$ row:

• $x_1$: $\frac{30}{170} = 0.176$
• $x_2$: $\frac{15}{170} = 0.088$
• $x_3$: $\frac{50}{170} = 0.294$

Step 3: Relative frequency marginal distribution (column-wise)

Now, we sum the relative frequencies for each column $x_1$, $x_2$, and $x_3$:

• $x_1$: $0.118 + 0.176 = 0.294$
• $x_2$: $0.147 + 0.088 = 0.235$
• $x_3$: $0.176 + 0.294 = 0.470$

Step 4: Verify the sum of all relative frequencies

The total sum of the relative frequencies should be close to 1 (due to rounding):

$0.294 + 0.235 + 0.470 = 0.999 \approx 1$

Final Answer:

The relative frequency marginal distribution, rounded to three decimal places, is:

	$x_1$	$x_2$	$x_3$
$y_1$	0.118	0.147	0.176
$y_2$	0.176	0.088	0.294
Total	0.294	0.235	0.470

Do you need further clarification or more details on any of the steps?

Related Questions:

1. How would you interpret a relative frequency marginal distribution in probability terms?
2. What is the difference between a relative frequency distribution and a frequency distribution?
3. Can you explain how marginal distributions are used in contingency tables?
4. How do you calculate the grand total in a two-way table for marginal distributions?
5. How would the relative frequency change if we doubled all the values in the table?

Tip:

Always check that the sum of all relative frequencies is close to 1 (or exactly 1, if there’s no rounding).