chapter 1

Chapter 3: Discrete Random Variables and Probability Distributions110. k(r,x) =( x + r −1)(x + r − 2)...( x + r − x)x!( 5.5)(4.5)(3.5)(2.5)44!2.5With r = 2.5 and p = .3, p(4) = (.3) (.7) = . 1068Using k(r,0) = 1, P(X ≥ 1) = 1 – p(0) = 1 – (.3) 2.5 = .9507111.a. p(x; λ,µ) = 1 1x;λ)p(x;µ )2p + where both p(x;λ) and p(x; µ) are Poisson p.m.f.’s(2and thus ≥ 0, so p(x; λ,µ) ≥ 0. Further,∞∑∞∞111 1p ( x;λ , µ ) = ∑ p(x;λ)+ ∑ p(x;µ ) = + = 1222 2x= 0x=0x=0b. . 6 p ( x;λ)+ .4 p(x;µ )∞∑11c. E(X) = x[p(x;λ)+ p(x;µ )] = xp(x;λ)+ xp(x;µ )=x=0 21 1 λ + µλ + µ =2 2 2212d. E(X 2 ) = x p(x;λ ) x p(x;µ ) = ( λ + λ)+ ( µ + µ ) (since for a∞∑x=012∞∑x=0∞∞1 2 1 2 1 2 1 2∑ + ∑222 2x=0x=0Poisson r.v., E(X 2 ) = V(X) + [E(X)] 2 = λ + λ 2 ),12so V(X) = [ λ + λ + µ + µ ]2222 λ − µ λ + µ⎡ λ + µ ⎤−⎢ 2 ⎥⎣ ⎦=⎛⎜⎝2⎞⎟⎠+2112.b(x + 1; n,p)b(x;n,p)( n − x)( x + 1)p(1 − p)a. = ⋅ > 1conclusion follows.if np – (1 – p) > x, from which the statedp(x + 1; λ)p(x;λ)λ( x + 1)b. = > 1 if x < λ - 1 , from which the stated conclusion follows. Ifλ is an integer, then λ - 1 is a mode, but p(λ,λ) = p(1 - λ, λ) so λ is also a mode[p(x; λ)]achieves its maximum for both x = λ - 1 and x = λ.127