The total lifetime (in year) of $5$ year old dog of a certain breed in a random variable whose distribution function is given by $f\left(x\right)=\left\{\begin{array}{ll}0& , \text{for} x\le 5\\ 1-\frac{25}{x^2},& \text{for} x>5\end{array}\right.$. Then the probability that such a five-year-old dog will live beyond $10$ years are

The probability density function of a continuous random variable $X$ is $f\left(x\right)=\left\{\begin{array}{ll}\frac{x}{8},& 0<x<4\\ 0,& \text{otherwise}\end{array}\right.$. Then the value of $P(X>3)$ is

If the probability density function of a continuous random variable $X$ is

$f\left(x\right)=\left\{\begin{array}{ll}K·e^{-\theta \mathrm{x}},& \theta >0, 0\le x<\infty \\ 0,& \text{otherwise}\end{array}\right.$. Then, $K=$

A random variable $X$ has the following probability distribution

For the probability distribution given by

The standard deviation $\left(\mathrm{\sigma }\right)$ is

A random variable $X$ has the following probability distribution:

Then, the expected value of $X$ is