If m is the parameter of the Poisson distribution then the variance of the distribution is

Variance =∑r=0∞r2p(r)−{∑r=0∞rp(r)}2 Now ∑r=0∞r2p(r)=12p(1)+22p(2)+32p(3)+............to    ∞ But r2p(r)=r2.e−mmrr=rr−4e−m.mr=(r−1)+1r−1e−m.mr                      =(1r−2+1r−1)e−m.mr=e−mm2mr−2r−2+e−m.m.mr−1r−1 ∴∑r2p(r)=e−m.n2∑mr−2r−2+e−mm.∑mr−1r−1                          =e−m.m2em+e−m.m.em=m2+m Again, ∑rp(r)=1.p(1)+2p(2)+3.p(3)+...........to   ∞ but rp(r)=r.e−m.mrr=e−m.mrr−1=e−m.mmr−1r−1 ∴∑rp(r)=e−mm.∑mr−1r−1=e−m.m.em=m ∴ Variance =(m2+m)−(m)2=m .