An online exam is attempted by $50$ candidates out of which $20$ are boys. The average marks obtained by boys is $12$ with a variance $2.$ The variance of marks obtained by $30$ girls is also $2.$ The average marks of all $50$ candidates is $15.$ If $\mu$ is the average marks of girls and ${\sigma }^2$ is the variance of marks of $50$ candidates, then $\mu +{\sigma }^2$ is equal to

The probability distribution of random variable $X$ is given by:

Let $p=P(1<X<4\mid X<3)$. If $5p=\lambda K$, then $\lambda$ is equal to

Consider the following frequency distribution:

If mean $=\frac{309}{22}$ and median $=14,$ then the value $(a-b{)}^2$ is equal to

Consider the following frequency distribution :

If the sum of all frequencies is $584$ and median is $45$, then $|\alpha -\beta |$ is equal to            .