Add up all the frequencies to the get the total number of occurrences:
2 + 2 + 16 + 10 + 9 + 4 + x + 2 = x + 45
Then the probability corresponding to a given frequency is equal to (frequency)/(sum of frequencies). For example, the probability that a randomly chosen student has read 1 book in the marathon is 2/(x + 45); the probability of having read 4 books is 10/(x + 45); etc.
The mean is computed by multiplying each book count by their respective probability, and adding these all up. If the mean is 4.28, then
[tex]1\cdot\dfrac2{x+45} + 2\cdot\dfrac2{x+45} + 3\cdot\dfrac{16}{x+45} + \cdots + 8\cdot\dfrac2{x+45} = 4.28[/tex]
Simplifying the left side gives
[tex]\dfrac{2+4+48+40+45+24+7x+16}{x+45} = 4.28 \\\\ \dfrac{7x+179}{x+45} = 4.28[/tex]
Solve for x :
[tex]7x+179 = 4.28(x+45) \\\\ 7x+179 = 4.28x + 192.6 \\\\ 2.72x = 13.6 \\\\ \boxed{x = 5}[/tex]
