TitleLet X be the binomial random variable. $($a$)$ Show that p X $1$k $+12=$ p X $1$ k $2$ n $-$ k p k $+11-$ p $2$DescriptionLet X be the binomial random variable.$($a$)$ Show thatp X $1$k $+12=$p X $1$ k $2$n $-$ k p k $+11-$ p$2$where p X$102=11-$ p n$.($b$)$ Show that part a implies that: $(1)$ P$3$X $=$ k$4$ is maximum at k max $=31$n $+12$p$4,$ where $[$ x $]$ denotes the largest integer that is smaller than or equal to x ; and $(2)$ when$1$n $+12$p is an integer, then the maximum is achieved at k max and k max $-1\mathrm{.}2$