Show that the distribution of Z may be approximated by a normal distribution for any large,

Show that the distribution of Z_λ may be approximated by a normal distribution for any large, not necessary integer, λ. (Advice: First, λ = [λ]+{λ}, where [λ] is the integer part of λ, and {λ}=λ−[λ], the fractional part of λ. Accordingly, with n = [λ], we may write Z_λ = Z_n + Y_{λ}, where Z_n and Y_{λ}, are independent Poisson r.v.’s with means n and {λ} respectively. Clearly, [λ]→∞ as λ→∞. To apply the CLT, one can switch to the normalized r.v.’s, and write

It remains to show that

(a) (Z_n −n)/√n is asymptotically normal;

(b ) [λ]/λ→ 1; and

(c) 1/λ(Y_{λ} −{λ}) ^P→0,

(d) These three facts considered together imply what we are proving.)