5 Quiz Exercises 2

Generating functions: Which of the following could be pgfs?
(i) $G_1(z)=3/(2+z);$ (ii) $G_2(z)=3/(2-z);$ (iii) $G_3(z)=2/(3-z)$.

• (a)

  Only (i);

• (b)

  Only (ii);

• (c)

  Only (iii);

• (d)

  More than one of the above;

• (e)

  None of the above.

Expectation and variance: Suppose $X$ is a random variable with expectation $\mu$, variance ${\sigma }^2$ and pgf $G_X(z)$. If $Y$ is a random variable with pgf $G_Y(z)=G_X{(z)}^3$, what are the expectation and variance of $Y$?

• (a)

  ${\mu }^3$ and ${\sigma }^6$;

• (b)

  $3\mu$ and $3{\sigma }^2+6{\mu }^2$;

• (c)

  $\mu$ and $3{\sigma }^2$;

• (d)

  $3\mu {\sigma }^2$ and $3{\sigma }^2-3{\mu }^2$;

• (e)

  None of the above.

Let $T$ be a random variable denoting the time at which three successes in a row first occur in Bernoulli trials with probability of success $p$; . Let $q=1-p$. The pgf of $T$ is given by

Suppose that $p=1/2$.

Probability 1: The probability that $T=3$ is

(a) 0;  (b) 1/2;  (c) $-1/2$;  (d) 1/8;  (e) None of the above.

Probability 2: The probability that $T=6$ is

(a) 1/8;  (b) 1/16;  (c) 1/32;  (d) 1/64;  (e) None of the above.

Expectation: The expected value of $T$ is

(a) 3;  (b) 7;  (c) 14;  (d) 20;  (e) None of the above.