Consider the random variable $X \sim {\text{Geo}}(p)$.

(a)     State ${\text{P}}(X < 4)$.

(b)     Show that the probability generating function for X is given by ${G_X}(t) = \frac{{pt}}{{1 - qt}}$, where $q = 1 - p$.

Let the random variable $Y = 2X$.

(c)     (i)     Show that the probability generating function for Y is given by ${G_Y}(t) = {G_X}({t^2})$.

          (ii)     By considering ${G'_Y}(1)$, show that ${\text{E}}(Y) = 2{\text{E}}(X)$.

Let the random variable $W = 2X + 1$.

(d)     (i)     Find the probability generating function for W in terms of the probability generating function of Y.

          (ii)     Hence, show that ${\text{E}}(W) = 2{\text{E}}(X) + 1$.

(a)     use of ${\text{P}}(X = n) = p{q^{n - 1}}{\text{ }}(q = 1 - p)$     (M1)

${\text{P}}(X < 4) = p + pq + p{q^2}{\text{ }}\left( { = 1 - {q^3}} \right){\text{ }}\left( { = 1 - {{(1 - p)}^3}} \right){\text{ }}( = 3p - 3{p^2} + {p^3})$     A1

[2 marks]

(b)   ${G_X}(t) = {\text{P}}(X = 1)t + {\text{P}}(X = 2){t^2} +  \ldots$     (M1)

$= pt + pq{t^2} + p{q^2}{t^3} +  \ldots$     A1

summing an infinite geometric series     M1

$= \frac{{pt}}{{1 - qt}}$     AG

[3 marks]

(c)     (i)     EITHER

          ${G_Y}(t) = {\text{P}}(Y = 1)t + {\text{P}}(Y = 2){t^2} +  \ldots$     A1

          $= 0 \times t + {\text{P}}(X = 1){t^2} + 0 \times {t^3} + {\text{P}}(X = 2){t^4} +  \ldots$     M1A1

          $= {G_X}({t^2})$     AG

          OR

          ${G_Y}(t) = E({t^Y}) = E({t^{2X}})$     M1A1

          $= E\left( {{{({t^2})}^X}} \right)$     A1

          $= {G_X}({t^2})$     AG

          (ii)     ${\text{E}}(Y) = {G'_Y}(1)$     A1

          EITHER

          $= 2t{G'_X}({t^2})$ evaluated at $t = 1$     M1A1

          $= 2{\text{E}}(X)$     AG

          OR

          $= \frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{p{t^2}}}{{(1 - q{t^2})}}} \right) = \frac{{2pt(1 - q{t^2}) + 2pq{t^3}}}{{{{(1 - q{t^2})}^2}}}$ evaluated at $t = 1$     A1

          $= 2 \times \frac{{p(1 - qt) + pqt}}{{{{(1 - qt)}^2}}}$ evaluated at $t = 1{\text{ (or }}\frac{2}{p})$     A1

          $= 2{\text{E}}(X)$     AG

[6 marks]

(d)     (i)     ${G_W}(t) = t{G_Y}(t)$ (or equivalent)     A2

          (ii)     attempt to evaluate ${G'_W}(t)$     M1

          EITHER

          obtain $1 \times {G_Y}(t) + t \times {G'_Y}(t)$     A1

          substitute $t = 1$ to obtain $1 \times 1 + 1 \times {G'_Y}(1)$     A1

          OR

          $= \frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{p{t^3}}}{{(1 - q{t^2})}}} \right) = \frac{{3p{t^2}(1 - q{t^2}) + 2pq{t^4}}}{{{{(1 - q{t^2})}^2}}}$     A1

          substitute $t = 1$ to obtain $1 + \frac{2}{p}$     A1

          $= 1 + 2{\text{E}}(X)$     AG

[5 marks]

Total [16 marks]