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 $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ We set $a = \frac{p \ln(p)}{p-1}$ and $x = r\ln(p) - a$. Prove that condition (II.2) is equivalent to the condition $$x \mathrm{e}^{x} \leqslant -\alpha a \mathrm{e}^{-a}.$$
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
$$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$
We set $a = \frac{p \ln(p)}{p-1}$ and $x = r\ln(p) - a$. Prove that condition (II.2) is equivalent to the condition
$$x \mathrm{e}^{x} \leqslant -\alpha a \mathrm{e}^{-a}.$$