Find cumulative distribution F(x)

1. The discrete random variable \(X\) has the probability distribution given in the table below.

1. Write down the value of \(\mathrm { F } ( 5 )\)
2. Find \(\mathrm { E } ( X )\)
3. Find \(\operatorname { Var } ( X )\) The random variable \(Y = 7 - 2 X\)
4. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
   3. \(\mathrm { P } ( Y > X )\) \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-03_2261_47_313_37}

1. The discrete random variable \(X\) can only take the values \(1,2,3\) and 4 For these values the cumulative distribution function is defined by

$$\mathrm { F } ( x ) = k x ^ { 2 } \text { for } x = 1,2,3,4$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability distribution of \(X\).

1. The discrete random variable \(X\) is defined by the cumulative distribution function

where \(k\) is a constant.

1. Find the probability distribution of \(X\).
2. Find \(\mathrm { P } ( 1.5 < X \leqslant 3.5 )\) The random variable \(Y = 12 - 7 X\)
3. Calculate Var(Y)
4. Calculate \(\mathrm { P } ( 4 X \leqslant | Y | )\)

3. When Rohit plays a game, the number of points he receives is given by the discrete random variable \(X\) with the following probability distribution.

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { F } ( 1.5 )\).
3. Show that \(\operatorname { Var } ( X ) = 1\)
4. Find \(\operatorname { Var } ( 5 - 3 X )\). Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
5. Find the probability that Rohit wins the prize.

1. The discrete random variable \(Y\) has the following probability distribution

where \(q , r\) and \(u\) are probabilities.

1. Write down the value of \(\mathrm { E } ( Y )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\) Given that \(F ( 0 ) = \frac { 19 } { 30 }\)
2. show that the value of \(u\) is \(\frac { 4 } { 15 }\) Given also that \(\operatorname { Var } ( Y ) = 37\)
3. find the value of \(q\) and the value of \(r\) The coordinates of a point \(P\) are \(( 12 , Y )\) The random variable \(D\) represents the length of \(O P\)
4. Find the probability distribution of \(D\)

7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.

1. Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
2. Write down \(\mathrm { F } ( 0.8 )\).
   1. Evaluate \(\operatorname { Var } ( X )\).

The discrete random variable \(X\) has the following probability distribution:

\(x\)	0	1	2	3	4	5
\(\text{P}(X = x)\)	0.11	0.17	0.2	0.13	\(p\)	\(p^2\)

1. Find the value of \(p\). [4 marks]
2. Find
   1. \(\text{P}(0 < X \leq 2)\),
   2. \(\text{P}(X \geq 3)\).
   [3 marks]
3. Find the mean and the variance of \(X\). [3 marks]
4. Construct a table to represent the cumulative distribution function \(\text{F}(x)\). [2 marks]