A random variable X has the following probability distribution :

The mean of the distribution is:

$\frac{19}{7}$

Given probability distribution

$X: 0,\;1,\;2,\;3,\;4$

$P(X): c,\;3c,\;4c,\;6c,\;7c$

Total probability = 1

$c+3c+4c+6c+7c=21c=1$

$c=\frac{1}{21}$

Mean of distribution

$E(X)=\sum xP(X)$

$E(X)=0\cdot c+1\cdot3c+2\cdot4c+3\cdot6c+4\cdot7c$

$=3c+8c+18c+28c=57c$

$E(X)=57\cdot\frac{1}{21}=\frac{57}{21}=\frac{19}{7}$

The mean of the distribution is $\frac{19}{7}$.