For the given probability distribution :

If mean $=\frac{19}{9}, $ then the value of constant A will be :

$\frac{1}{3}$

The correct answer is Option (3) → $\frac{1}{3}$

Since, sum of all probabilities must be 1.

$A+\frac{2}{3}A+\frac{4}{27}+\frac{8A}{9}=1$

$⇒A\left(\frac{9+6+8}{9}\right)=1-\frac{4}{27}=\frac{23}{27}$

$⇒A×\frac{23}{9}=\frac{23}{27}$

$⇒A=\frac{9}{27}=\frac{1}{3}$