Multiple observations or trials

3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,3 ]\). Find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. \(\mathrm { P } ( X < 1.4 )\) A total of 40 observations of \(X\) are made.
5. Find the probability that at least 10 of these observations are negative.

2. The continuous random variable \(X\) represents the error, in mm, made when a machine cuts piping to a target length. The distribution of \(X\) is rectangular over the interval \([ - 5.0,5.0 ]\). Find

1. \(\mathrm { P } ( X < - 4.2 )\),
2. \(\mathrm { P } ( | X | < 1.5 )\). A supervisor checks a random sample of 10 lengths of piping cut by the machine.
3. Find the probability that more than half of them are within 1.5 cm of the target length.
   (3 marks)
   If \(X < - 4.2\), the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
4. Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
   (5 marks)

5. In a party game, a bottle is spun and whoever it points to when it stops has to play next. The acute angle, in degrees, that the bottle makes with the side of the room is modelled by a rectangular distribution over the interval [0,90]. Find the probability that on one spin this angle is

1. between \(25 ^ { \circ }\) and \(38 ^ { \circ }\),
2. \(45 ^ { \circ }\) to the nearest degree. The bottle is spun ten times.
3. Find the probability that the acute angle it makes with the side of the room is less than \(10 ^ { \circ }\) more than twice.

8 The random variable \(X\) has a continuous uniform distribution over [0,10].

1. Find the probability that, if two independent values of \(X\) are taken, one is less than 3 and the other is greater than 3 . The random variable \(T\) denotes the sum of 5 independent values of \(X\).
2. State the value of \(\mathrm { P } ( T \leqslant 25 )\). The spreadsheet below shows the heading row and the first 20 data rows from a total of 100 data rows of a simulation of the distribution of \(X\). Each of the 100 rows shows a simulation of 5 independent values of \(X\), together with \(T\), the sum of the 5 values. All of the values have been rounded to 2 decimal places. In column I the spreadsheet shows the number of values of \(T\) that are less than or equal to the corresponding values in column H . For example, there are 75 simulated values of \(T\) that are less than or equal to 30 .

   	A	B	c	D	E	F	G	H	I
   1	\(\mathrm { X } _ { 1 }\)	\(\mathrm { X } _ { 2 }\)	\(\mathrm { X } _ { 3 }\)	\(\mathrm { X } _ { 4 }\)	\(\mathrm { X } _ { 5 }\)	T		t	Number \(\leqslant \mathrm { t }\)
   2	3.73	6.65	4.93	0.41	9.33	25.06		0	0
   3	4.95	6.58	4.48	2.51	7.26	25.79		5	0
   4	8.10	4.87	4.26	3.83	0.79	21.85		10	1
   5	6.70	4.10	5.10	1.82	6.76	24.48		15	4
   6	3.73	8.38	8.49	9.87	1.31	31.79		20	23
   7	3.22	4.36	0.12	1.34	9.49	18.53		25	48
   8	9.17	7.13	5.47	4.35	2.44	28.55		30	75
   9	3.42	1.93	6.04	2.99	8.85	23.24		35	93
   10	0.98	0.68	9.82	9.83	7.28	28.58		40	99
   11	5.86	1.67	7.77	4.08	7.14	26.52		45	100
   12	9.20	0.31	5.82	5.31	6.45	27.10		50	100
   13	7.04	4.30	2.06	0.06	4.16	17.62
   14	0.31	5.02	1.48	5.37	1.77	13.94
   15	3.77	6.04	1.21	7.67	5.01	23.69
   16	1.21	5.54	1.90	1.43	6.91	17.00
   17	9.27	1.98	5.80	9.37	9.34	35.76
   18	4.30	5.66	2.80	1.56	1.19	15.51
   19	7.15	3.19	6.89	5.41	2.18	24.82
   20	6.18	6.32	3.01	6.49	9.12	31.13
   21	5.03	5.99	5.19	6.97	3.55	26.73

3. Use the spreadsheet output to estimate each of the following.
   • \(\mathrm { P } ( T \leqslant 25 )\)
   • \(\mathrm { P } ( T > 35 )\)
   • In this question you must show detailed reasoning.
   The random variable \(Y\) is the mean of 100 independent values of \(T\). Determine an estimate of \(\mathrm { P } ( Y > 26 )\).