A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ With $a = \frac{p\ln(p)}{p-1}$ and $x = r\ln(p) - a$, condition (II.2) is equivalent to $xe^x \leqslant -\alpha a e^{-a}$. Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, study the existence of a largest natural integer $r$ satisfying condition (II.2).
A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition
$$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$
With $a = \frac{p\ln(p)}{p-1}$ and $x = r\ln(p) - a$, condition (II.2) is equivalent to $xe^x \leqslant -\alpha a e^{-a}$. Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, study the existence of a largest natural integer $r$ satisfying condition (II.2).