Multiple unknowns from expectation and variance

2 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).

1. The discrete random variable \(X\) can take only the values \(1,2,3\) and 4 . For these values, the probability function is given by

$$\mathrm { P } ( X = x ) = \frac { a x + b } { 60 } \quad \text { for } x = 1,2,3,4$$ where \(a\) and \(b\) are constants.

1. Show that \(5 a + 2 b = 30\) Given that \(\mathrm { F } ( 3 ) = \frac { 13 } { 20 }\)
2. find the value of \(a\) and the value of \(b\) Given also that \(Y = X ^ { 2 }\)
3. find the cumulative distribution function of \(Y\)

4. The discrete random variable \(X\) has probability distribution

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\) For this probability distribution, \(\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)\)
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find the value of \(a\) and the value of \(b\)
3. Find \(\operatorname { Var } ( 1 - 3 X )\) Given that \(Y = 1 - X\), find
4. 1. \(\mathrm { P } ( Y < 0 )\)
   2. the largest possible value of \(k\) such that \(\mathrm { P } ( Y < k ) = 0.2\)

1. The probability distribution of the discrete random variable \(X\) is given by

where \(a\) is a constant.

1. Find, in terms of \(a , \mathrm { E } ( X )\)
2. Find the range of the possible values of \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 0.56\)
3. find the possible values of \(a\)

5. The discrete random variable \(X\) has the following probability distribution where \(a\), \(b\) and \(c\) are probabilities.
Given that \(\mathrm { E } ( X ) = 0.8\)

1. find the value of \(c\). Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 5\) find
2. the value of \(a\) and the value of \(b\),
3. \(\operatorname { Var } ( X )\) The random variable \(Y = 5 - 3 X\)
   Find
4. \(\mathrm { E } ( Y )\)
5. \(\operatorname { Var } ( Y )\)
6. \(\mathrm { P } ( Y \geqslant 0 )\)

1. In a game, a player can score \(0,1,2,3\) or 4 points each time the game is played.

The random variable \(S\), representing the player's score, has the following probability distribution where \(a , b\) and \(c\) are constants. The probability of scoring less than 2 points is twice the probability of scoring at least 2 points. Each game played is independent of previous games played.
John plays the game twice and adds the two scores together to get a total.
Calculate the probability that the total is 6 points.

3 The discrete random variable \(R\) has the following probability distribution. It is known that \(\mathrm { E } ( R ) = 0.2\) and \(\operatorname { Var } ( R ) = 3.56\)
Find the values of \(a , b\) and \(c\).
[0pt] [4 marks]

3 A discrete random variable \(X\) has the following probability distribution.

1. Determine the value of \(p\).
2. It is given that \(\mathrm { E } ( a X + b ) = \operatorname { Var } ( a X + b ) = 23.19\), where \(a\) and \(b\) are positive constants. Determine the value of \(a\) and the value of \(b\).

1 A discrete random variable \(X\) has the following distribution, where \(a , b\) and \(c\) are constants. It is given that \(\mathrm { E } ( X ) = 1.25\) and \(\operatorname { Var } ( X ) = 0.8875\).

1. Determine the values of \(a\), \(b\) and \(c\).
2. The random variable \(Y\) is defined by \(Y = 7 - 2 X\). Write down the value of \(\operatorname { Var } ( Y )\).
3. Twenty independent observations of \(X\) are obtained. The number of those observations for which \(X = 3\) is denoted by \(T\). Find the value of \(\operatorname { Var } ( T )\).

1. The discrete random variable \(X\) has the following probability distribution

Given that \(\mathrm { E } ( X ) = 0.5\)

1. find the value of \(a\). Given also that \(\operatorname { Var } ( X ) = 5.01\)
2. find the value of \(b\) and the value of \(c\). The random variable \(Y = 5 - 8 X\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)

5. The score when a spinner is spun is given by the discrete random variable \(X\) with the following probability distribution, where \(a\) and \(b\) are probabilities.

1. Explain why \(\mathrm { E } ( X ) = 2\)
2. Find a linear equation in \(a\) and \(b\). Given that \(\operatorname { Var } ( X ) = 7.1\)
3. find a second equation in \(a\) and \(b\) and simplify your answer.
4. Solve your two equations to find the value of \(a\) and the value of \(b\). The discrete random variable \(Y = 10 - 3 X\)
5. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\) The spinner is spun once.
6. Find \(\mathrm { P } ( Y > X )\).

6 The probability distribution of a discrete random variable, \(X\), is shown in the table below.

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\).
2. 1. In the case where \(\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4\), find an expression for \(\operatorname { Var } ( X )\) in terms of \(a\).
   2. In this case, show that the greatest possible value of \(\operatorname { Var } ( X )\) is 0.65 . You must state the associated value of \(a\).
3. You are now given instead that \(\mathrm { E } ( X )\) is not known.
   1. State the least possible value of \(\operatorname { Var } ( X )\).
   2. Give all possible pairs of values of \(a\) and \(b\) which give the least possible value of \(\operatorname { Var } ( X )\) stated in part (c)(i).

3 In this question you must show detailed reasoning. A discrete random variable \(V\) has the following probability distribution, where \(p\) and \(q\) are constants. It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).