Multiple unknowns from expectation and variance

Questions providing a partial probability distribution with multiple unknown constants and asking to find them using constraints including both a given expectation and a given variance.

16 questions · Standard +0.3

5.02b Expectation and variance: discrete random variables
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CAIE S1 2012 November Q2
6 marks Standard +0.8
2 The discrete random variable \(X\) has the following probability distribution.
\(x\)- 3024
\(\mathrm { P } ( X = x )\)\(p\)\(q\)\(r\)0.4
Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).
Edexcel S1 2017 January Q7
10 marks Standard +0.3
  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 \(Y\)
Edexcel S1 2018 January Q4
13 marks Standard +0.3
4. The discrete random variable \(X\) has probability distribution
\(x\)- 4- 3125
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(a\)\(b\)0.2
  1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\) For this probability distribution, \(\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)\)
    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
    1. \(\mathrm { P } ( Y < 0 )\)
    2. the largest possible value of \(k\) such that \(\mathrm { P } ( Y < k ) = 0.2\)
Edexcel S1 2023 January Q3
11 marks Standard +0.3
  1. The probability distribution of the discrete random variable \(X\) is given by
\(x\)234
\(\mathrm { P } ( X = x )\)\(a\)0.4\(0.6 - a\)
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 \(a\)
Edexcel S1 2014 June Q5
15 marks Standard +0.3
5. The discrete random variable \(X\) has the following probability distribution
\(x\)- 2024
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(a\)\(c\)
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. \(\mathrm { P } ( Y \geqslant 0 )\)
Edexcel AS Paper 2 2020 June Q3
6 marks Standard +0.8
  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.
\(s\)01234
\(\mathrm { P } ( S = s )\)\(a\)\(b\)\(c\)0.10.15
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.
AQA Further AS Paper 2 Statistics Specimen Q3
4 marks Standard +0.3
3 The discrete random variable \(R\) has the following probability distribution.
\(\boldsymbol { r }\)- 20\(a\)4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)0.3\(b\)\(c\)0.1
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]
OCR Further Statistics AS 2022 June Q3
9 marks Standard +0.8
3 A discrete random variable \(X\) has the following probability distribution.
\(x\)1234
\(\mathrm { P } ( X = x )\)\(p\)0.310.3\(p ^ { 2 }\)
  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 \(b\).
OCR Further Statistics 2024 June Q1
8 marks Standard +0.3
1 A discrete random variable \(X\) has the following distribution, where \(a , b\) and \(c\) are constants.
\(x\)0123
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)\(a\)\(b\)\(c\)0.1
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 \(\operatorname { Var } ( T )\).
Edexcel S1 2021 June Q5
15 marks Standard +0.3
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)- 2- 1014
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(c\)\(b\)\(a\)
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 \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)
Edexcel S1 2018 June Q5
14 marks Standard +0.3
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.
\(x\)- 10245
\(\mathrm { P } ( X = x )\)\(b\)\(a\)\(a\)\(a\)\(b\)
  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 \(\mathrm { P } ( Y > X )\).
OCR MEI Further Statistics Minor 2024 June Q6
9 marks Standard +0.8
6 The probability distribution of a discrete random variable, \(X\), is shown in the table below.
\(x\)012
\(\mathrm { P } ( X = x )\)\(1 - a - b\)\(a\)\(b\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\).
    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).
OCR Further Statistics 2022 June Q3
8 marks Standard +0.8
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.
\(v\)0123
\(\mathrm { P } ( \mathrm { V } = \mathrm { v } )\)\(p\)\(q\)0.120.2
It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).
CAIE S1 2023 March Q2
7 marks Moderate -0.3
Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6. The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.
  1. Show that the probability of obtaining exactly one head is 0.225. [3]
  2. Complete the following probability distribution table for \(X\). [2]
    \(x\)01234
    P(\(X = x\))0.050.2250.075
  3. Given that E(\(X\)) = 2.1, find the value of Var(\(X\)). [2]
Edexcel S1 2002 November Q6
15 marks Moderate -0.8
The discrete random variable \(X\) has the following probability distribution.
\(x\)\(-2\)\(-1\)\(0\)\(1\)\(2\)
\(P(X = x)\)\(\alpha\)\(0.2\)\(0.1\)\(0.2\)\(\beta\)
  1. Given that \(E(X) = -0.2\), find the value of \(\alpha\) and the value of \(\beta\). [6]
  2. Write down \(F(0.8)\). [1]
  3. Evaluate \(\text{Var}(X)\). [4]
Find the value of
  1. \(E(3X - 2)\), [2]
  2. \(\text{Var}(2X + 6)\). [2]
OCR FS1 AS 2021 June Q2
8 marks Standard +0.3
The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(P(W = w)\)0.250.36\(x\)\(x^2\)
  1. Show that \(\text{Var}(W) = 0.8571\). [7]
  2. Find \(\text{Var}(3W + 6)\). [1]