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 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}

A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{30}\) [2]

Find the exact value of

1. P\((1 < Y \leqslant 4)\) [2]
2. E\((Y)\) [2]

The random variable \(X = 15 - 2Y\)

1. Calculate P\((Y \geqslant X)\) [3]
2. Calculate Var\((X)\) [4]

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

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

Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for \(N\), the number of checkouts that are staffed at any given time during the spring, is $$P(N = n) = \begin{cases} \frac{3}{4}\left(\frac{1}{4}\right)^{n-1} & \text{for } n = 1, 2 \\ k & \text{for } n = 3 \end{cases}$$

1. Find the value of \(k\). [1 mark]
2. Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed. [2 marks]