The Probability distribution of a random variable X is given as :

The mean of the distribution is :

$\frac{53}{19}$

The correct answer is Option (4) → $\frac{53}{19}$

The sum of all probabilities,

$k+2k+4k+5k+7k=1$

$19k=1$

$k=\frac{1}{19}$

The mean or expected value is,

$E(X)=\sum\limits_{x}P(X=x)x$

$=0+\frac{2}{19}+\frac{8}{19}+\frac{15}{19}+\frac{28}{19}$

$=\frac{53}{19}$