Probability Generating Functions (Edexcel A-Level Further Mathematics): Quizzes

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If XX and YY are independent, what is GZ(t)G_Z(t) for Z=X+YZ = X + Y?

GZ(t)=GX(t)×GY(t)G_Z(t) = G_X(t) \times G_Y(t)

If Z=aXZ = aX, what is the PGF of ZZ?

GZ(t)=GX(ta)G_Z(t) = G_X(t^a)

If Z=X+cZ = X + c where cc is a constant, what is GZ(t)G_Z(t)?

GZ(t)=tcGX(t)G_Z(t) = t^c G_X(t)

For Z=X+YZ = X + Y, what is E[Z]\mathbb{E}[Z]?

E[X]+E[Y]\mathbb{E}[X] + \mathbb{E}[Y]

For independent XX and YY, what is Var(X+Y)\text{Var}(X + Y)?

Var(X)+Var(Y)\text{Var}(X) + \text{Var}(Y)

If X∼Po(3)X \sim \text{Po}(3) and Y∼Po(5)Y \sim \text{Po}(5) are independent, what is X+YX + Y?

Po(8)\text{Po}(8)

For X∼B(5,0.4)X \sim B(5, 0.4) and Y∼B(7,0.4)Y \sim B(7, 0.4) independent, what is E[X+Y]\mathbb{E}[X + Y]?

4.84.8

If X∼Geom(0.5)X \sim \text{Geom}(0.5) with GX(t)=0.51−0.5tG_X(t) = \frac{0.5}{1-0.5t}, what is GZ(t)G_Z(t) for Z=2XZ = 2X?

0.51−0.5t2\frac{0.5}{1-0.5t^2}

For Sn=X1+X2+⋯+XnS_n = X_1 + X_2 + \dots + X_n (independent), what is GSn(t)G_{S_n}(t)?

∏i=1nGXi(t)\prod_{i=1}^n G_{X_i}(t)

When does GZ(t)=GX(t)×GY(t)G_Z(t) = G_X(t) \times G_Y(t) apply for Z=X+YZ = X + Y?

When XX and YY are independent

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