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.

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. \(0.40 \quad 0.42 \quad 0.58 \quad 0.66\)

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

The random variable R has the binomial distribution B(12, 0.35).

1. Find P(R ≥ 4). [2]

The random variable S has the Poisson distribution with mean 2.71.

1. Find P(S ≤ 1). [3]

The random variable T has the normal distribution N(2.5, 5²).

1. Find P(T ≤ 18). [2]

The random variable \(R\) has the binomial distribution B(12, 0.35).

1. Find P(\(R \geq 4\)). [2]

The random variable \(S\) has the Poisson distribution with mean 2.71.

1. Find P(\(S \leq 1\)). [3]

The random variable \(T\) has the normal distribution N(25, \(5^2\)).

1. Find P(\(T \leq 18\)). [2]

The discrete random variable \(X\) has probability distribution The discrete random variable \(Y\) has probability distribution It is claimed that P(X ≥ 3) is greater than P(Y ≤ 4) Determine if this claim is correct. Fully justify your answer. [4 marks]

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 \(P(3 \leq X < 6)\) Circle the correct answer. [1 mark] 0.26 \quad\quad 0.31 \quad\quad 0.35 \quad\quad 0.40