Variance of geometric distribution

6 A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by \(p\).

1. Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of \(p\), for the variance of the number of green beads Sasha selects. Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the \(X\) th bead that Freda selects.
2. Assume that \(p = 0.3\). Find
   1. \(\mathrm { P } ( X \geqslant 5 )\),
   2. \(\operatorname { Var } ( X )\).
3. In fact, on the basis of a large number of observations of \(X\), it is found that \(\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )\). Estimate the value of \(p\).

5 The discrete random variable \(X\) has a geometric distribution. It is given that \(\operatorname { Var } ( X ) = 20\).
Determine \(\mathrm { P } ( X \geqslant 7 )\).

7 The random variable \(D\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\operatorname { Var } ( D ) = \frac { 40 } { 9 }\).
Determine

1. \(\operatorname { Var } ( 3 D + 5 )\),
2. \(\mathrm { E } ( 3 \mathrm { D } + 5 )\),
3. \(\mathrm { P } ( D > \mathrm { E } ( D ) )\).

A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac{1}{3}\) that it lands on blue. The spinner is spun repeatedly. The random variable \(B\) represents the number of the spin when the spinner first lands on blue.

1. Find
   1. P(\(B = 4\))
   2. P(\(B \leq 5\))
   [4]
2. Find E(\(B^2\)) [3]

Steve invites Tamara to play a game with this spinner. Tamara must choose a colour, either red or blue. Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(e^X\) If Tamara chooses blue, her score is \(X^2\)

1. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]

The random variable \(D\) has the distribution Geo\((p)\). It is given that Var\((D) = \frac{40}{9}\). Determine

1. Var\((3D + 5)\). [1]
2. E\((3D + 5)\). [6]
3. \(\text{P}(D > \text{E}(D))\). [3]