Let $$f(x)$$ be the continuous probability density function of a random variable X. The probability that $$a\, < \,X\, \le \,b$$, is:

The following is the Hasse diagram of the poset $$\left[ {\left\{ {a,b,c,d,e} \right\}, \prec } \right]$$

The poset is:

The set $$\left\{ {1,\,\,2,\,\,4,\,\,7,\,\,8,\,\,11,\,\,13,\,\,14} \right\}$$ is a group under multiplication modulo $$15$$. The inverse of $$4$$ and $$7$$ are respectively:

An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the tails are independent, the expected number of tosses are