# For $X \sim \mathrm{Binomial}(n,\frac{1}{2})$ does there exist $a,b,c,Y$ s.t. $\Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$?

I need to upper bound some complicated expressions involving binomial distributions:

Let $X \sim \mathrm{Binomial}(n,\frac{1}{2})$.

I want to find $a,b,c,m$ such that for $Y \sim \mathrm{Binomial}(m,\frac{1}{2})$.

$\forall x \in \{0,\ldots,n\} : \Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$

and this upper bound is tight in some sense. I am loose in the definition of tight on purpose, since I am not completely aware of what I can hope for.

Does there exist general results about problems like this?

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