Calculate basic probabilities

2 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = x ) = \begin{cases} 0.1 & x = 0,1,2,3,4,5,6,7,8,9
0 & \text { otherwise } \end{cases}$$ Find the value of \(\mathrm { P } ( 4 \leq X \leq 7 )\)
Circle your answer.

3. The random variable \(X\) has the discrete uniform distribution over the set of consecutive integers \(\{ - 7 , - 6 , \ldots , 10 \}\).
Calculate (a) the expectation and variance of \(X\),
(b) \(\mathrm { P } ( X > 7 )\),
(c) the value of \(n\) for which \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).

4. A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13 . A card is selected at random. The random variable \(X\) represents the number on the card.

1. Find \(\mathrm { P } ( X \leq 5 )\).
2. Name the distribution of \(X\) and find the expectation and variance of \(X\). A hand of 12 cards consists of three 2 s , four 3 s , two 4 s , two 5 s and one 6 . The random variable \(Y\) represents the number on a card chosen at random from this hand.
3. Draw up a table to show the probability distribution of \(Y\).
4. Calculate \(\operatorname { Var } ( 3 Y - 2 )\). \section*{STATISTICS 1 (A) TEST PAPER 8 Page 2}

6 A lottery has tickets numbered 1 to \(n\) inclusive, where \(n\) is a positive integer. The random variable \(X\) denotes the number on a ticket drawn at random.

1. Determine \(\mathrm { P } \left( \mathrm { X } \leqslant \frac { 1 } { 4 } \mathrm { n } \right)\) in each of the following cases.
   1. \(n\) is a multiple of 4 .
   2. \(n\) is of the form \(4 k + 1\), where \(k\) is a positive integer. Give your answer as a single fraction in terms of \(n\).
2. Given that \(n = 101\), find the probability that \(X\) is within one standard deviation of the mean.

1 The discrete random variable \(X\) has the following probability distribution function. $$\mathrm { P } ( X = x ) = \begin{cases} 0.2 & x = 1
0.3 & x = 2
0.1 & x = 3,4
0.25 & x = 5
0.05 & x = 6
0 & \text { otherwise } \end{cases}$$ Find the mode of \(X\). Circle your answer.
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0.10 .2523
\(2 \quad \mathrm {~A} \chi ^ { 2 }\) test is carried out in a school to test for association between the class a student belongs to and the number of times they are late to school in a week. The contingency table below gives the expected values for the test. Find a possible value for the degrees of freedom for the test. Circle your answer. 681215

1 The discrete uniform distribution \(X\) can take values \(1,2,3 , \ldots , 10\)
Find \(\mathrm { P } ( X \geq 7 )\) Circle your answer. \(0.3 \quad 0.4 \quad 0.6 \quad 0.7\)