Consider a Gaussian distributed random variable with zero mean and standard deviation $$\sigma .\,\,\,$$ The value of its cumulative distribution function at the origin will be

A random variable is uniformly distributed over the interval $$2$$ to $$10.$$ Its variance will be

$${P_x}\left( X \right) = M{e^{\left( { - 2\left| x \right|} \right)}} + N{e^{\left( { - 3\left| x \right|} \right)}}\,\,$$ is the probability density function for the real random variable $$X,$$ over the entire $$x$$-axis, $$M$$ and $$N$$ are both positive real numbers. The equation relating $$M$$ and $$N$$ is

Consider the differential equation $${{dy} \over {dx}} = 1 + {y^2}.$$ Which one of the following can be particular solution of this differential equation ?