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 \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).

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