Direct probability from given distribution

3 The possible values of the random variable \(X\) are the 8 integers in the set \(\{ - 2 , - 1,0,1,2,3,4,5 \}\). The probability of \(X\) being 0 is \(\frac { 1 } { 10 }\). The probabilities for all the other values of \(X\) are equal. Calculate

1. \(\mathrm { P } ( X < 2 )\),
2. the variance of \(X\),
3. the value of \(a\) for which \(\mathrm { P } ( - a \leqslant X \leqslant 2 a ) = \frac { 17 } { 35 }\).

2. A spinner can land on the numbers \(2,4,5,7\) or 8 only. The random variable \(X\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(X\) is given in the table below.

1. Find \(\mathrm { P } ( 2 X - 3 > 5 )\) Given that \(\mathrm { E } ( X ) = 4.6\)
2. show that \(\operatorname { Var } ( X ) = 4.14\) The random variable \(Y = a X - b\) where \(a\) and \(b\) are positive constants.
   Given that $$\mathrm { E } ( Y ) = 13.4 \quad \text { and } \quad \operatorname { Var } ( Y ) = 66.24$$
3. find the value of \(a\) and the value of \(b\) In a game Sam and Alex each spin the spinner once, landing on \(X _ { 1 }\) and \(X _ { 2 }\) respectively.
   Sam's score is given by the random variable \(S = X _ { 1 }\)
   Alex's score is given by the random variable \(R = 2 X _ { 2 } - 3\)
   The person with the higher score wins the game. If the scores are the same it is a draw.
4. Find the probability that Sam wins the game.

3 Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.

1. Find the probability that a member borrows more than 3 books.
2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books.
3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\).
4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\).
5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
   1. the mean amount that will be paid by a member;
   2. the standard deviation of the amount that will be paid by a member.

13 The diagram below shows the probability distribution for a discrete random variable \(Y\).
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-17_816_1338_356_351} Find \(\mathrm { P } ( 0 < Y \leq 3 )\).
Circle your answer. \(0.40 \quad 0.42 \quad 0.58 \quad 0.66\)

13 The number of pots of yoghurt, \(X\), consumed per week by adults in Milton is a discrete random variable with probability distribution given by Find \(\mathrm { P } ( 3 \leq X < 6 )\) Circle the correct answer. \(0.26 \quad 0.31 \quad 0.35 \quad 0.40\)

3 The discrete random variable \(X\) has the following probability distribution The continuous random variable \(Y\) has the following probability density function $$\mathrm { f } ( y ) = \begin{cases} \frac { 1 } { 64 } y ^ { 3 } & 0 \leq y \leq 4
0 & \text { otherwise } \end{cases}$$ Given that \(X\) and \(Y\) are independent, show that \(\mathrm { E } \left( X ^ { 2 } + Y ^ { 2 } \right) = \frac { 1327 } { 60 }\)

1 The discrete random variable \(X\) has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 5 - x } { 10 } & x = 1,2,3,4
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\)
Circle your answer.
0.1
0.15
0.2
0.3

1 The discrete random variable \(X\) has the following probability distribution Find \(\mathrm { P } ( X > 18 )\)
Circle your answer.
0.1
0.2
0.7
0.8