Piecewise or conditional probability function

5. The random variable \(X\) has probability function $$P ( X = x ) = \begin{cases} k x , & x = 1,2,3
k ( x + 1 ) , & x = 4,5 \end{cases}$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the exact value of \(\mathrm { E } ( X )\).
3. Show that, to 3 significant figures, \(\operatorname { Var } ( X ) = 1.47\).
4. Find, to 1 decimal place, \(\operatorname { Var } ( 4 - 3 X )\).

1. A discrete random variable \(X\) has the probability function

$$\mathrm { P } ( X = x ) = \begin{cases} k ( 1 - x ) ^ { 2 } & x = - 1,0,1 \text { and } 2
0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 1 } { 6 }\)
2. Find \(\mathrm { E } ( X )\)
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 3 }\)
4. Find \(\operatorname { Var } ( 1 - 3 X )\)

5. The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 2,4,6
k ( x - 2 ) & x = 8
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 18 }\)
2. Find the exact value of \(\mathrm { F } ( 5 )\).
3. Find the exact value of \(\mathrm { E } ( X )\).
4. Find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\).
5. Calculate \(\operatorname { Var } ( 3 - 4 X )\) giving your answer to 3 significant figures.

6. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) & x = 0,1,2
k ( x - 2 ) & x = 3
0 & \text { otherwise } \end{cases}\)
where \(k\) is a positive constant.
a Show that \(k = 0.25\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
b Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\)
c Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
d Find \(\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)\)

1. Answer all the questions.

The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where \(c\) and \(k\) are constants.

1. Show that \(16 c + k = 1\)
2. Given that \(\mathrm { P } ( X \geq 3 ) = \frac { 5 } { 8 }\) find the value of \(c\) and the value of \(k\).
   [0pt] [BLANK PAGE]

7. The probability distribution of a random variable \(X\) is modelled as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 ,
0 & \text { otherwise, } \end{cases}\)
where \(k\) is a constant.

1. Show that \(k = \frac { 12 } { 25 }\).
2. Show in a table the values of \(X\) and their probabilities.
3. The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\). Find \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)\). In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7 .
4. Determine the probability that a total of exactly 7 is first reached on the 5th observation. END OF TEST

6 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for \(R\), the number of checkouts that are staffed at any given time, is $$\mathrm { P } ( R = r ) = \left\{ \begin{array} { c l } \frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { r - 1 } & r = 1,2,3
k & r = 4 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 27 }\).
2. Find the probability that, at any given time, there will be at least 3 checkouts that are staffed.
3. It is suggested that the total number of customers, \(C\), that can be served at the checkouts per hour may be modelled by $$C = 27 R + 5$$ Find:
   1. \(\mathrm { E } ( C )\);
   2. the standard deviation of \(C\).

15 The discrete random variable \(X\) is modelled by the probability distribution defined by: $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c x & x = 1,2
k x ^ { 2 } & x = 3,4
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(c\) are constants.
15

1. State, in terms of \(k\), the probability that \(X = 3\)
   15
2. Given that \(\mathrm { P } ( X \geq 3 ) = 3 \times \mathrm { P } ( X \leq 2 )\)
   Find the exact value of \(k\) and the exact value of \(c\).
   \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-21_2488_1716_219_153}