Probability distributions with parameters

Questions providing a discrete probability distribution with unknown constants/parameters and asking to find these constants using given conditions like E(X), Var(X), or normalization.

9 questions

OCR S4 2010 June Q6
6 Nuts and raisins occur in randomly chosen squares of a particular brand of chocolate. The numbers of nuts and raisins are denoted by \(N\) and \(R\) respectively and the joint probability distribution of \(N\) and \(R\) is given by $$f ( n , r ) = \begin{cases} c ( n + 2 r ) & n = 0,1,2 \text { and } r = 0,1,2
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.
  1. Find the value of \(c\).
  2. Find the probability that there is exactly one nut in a randomly chosen square.
  3. Find the probability that the total number of nuts and raisins in a randomly chosen square is more than 2 .
  4. For squares in which there are 2 raisins, find the mean number of nuts.
  5. Determine whether \(N\) and \(R\) are independent.
Edexcel FS1 AS 2018 June Q3
  1. A fair six-sided black die has faces numbered \(1,2,2,3,3\) and 4
The random variable \(B\) represents the score when the black die is rolled.
  1. Write down the value of \(\mathrm { E } ( B )\) A white die has 6 faces numbered \(1,1,2,4,5\) and \(c\) where \(c > 5\)
    The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by
    \(w\)1245\(c\)
    \(\mathrm { P } ( W = w )\)\(a + b\)\(a\)0.3\(a\)\(b\)
    Greg and Nilaya play a game with these dice.
    Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
    The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
  2. Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\)
    Given that \(\mathrm { E } ( X ) = 2.6\)
  3. find the exact value of \(\operatorname { Var } ( X )\)
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FS1 AS 2019 June Q4
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 3- 1124
\(\mathrm { P } ( X = x )\)\(q\)\(\frac { 7 } { 30 }\)\(\frac { 7 } { 30 }\)\(q\)\(r\)
where \(q\) and \(r\) are probabilities.
  1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
  2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
  3. find the value of \(q\) and the value of \(r\)
  4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)
Edexcel FS1 AS 2020 June Q3
  1. The probability distribution of the discrete random variable \(X\) is
$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3
\frac { m } { 2 x } & \text { for } x = 6 \text { and } 9
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)
Edexcel FS1 AS 2022 June Q4
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)0236
\(\mathrm { P } ( X = x )\)\(p\)0.25\(q\)0.4
  1. Find in terms of \(q\)
    1. \(\mathrm { E } ( X )\)
    2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
  2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\)
    Given that games are independent,
  3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
  4. Find the probability that Bobby wins at least \(\pounds 10\)
Edexcel FS1 AS 2023 June Q1
  1. The discrete random variable \(X\) has the following distribution
\(x\)01234
\(\mathrm { P } ( X = x )\)\(r\)\(k\)\(\frac { k } { 2 }\)\(\frac { k } { 3 }\)\(\frac { k } { 4 }\)
where \(r\) and \(k\) are positive constants.
The standard deviation of \(X\) equals the mean of \(X\)
Find the exact value of \(r\)
Edexcel FS1 AS 2024 June Q3
  1. The discrete random variable \(X\) has probability distribution,
\(x\)- 10137
\(\mathrm { P } ( X = x )\)\(p\)\(r\)\(p\)0.3\(r\)
where \(p\) and \(r\) are probabilities.
Given that \(\mathrm { E } ( X ) = 1.95\)
find the exact value of \(\mathrm { E } ( \sqrt { X + 1 } )\) giving your answer in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
(6)
Edexcel FS1 2022 June Q2
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 5- 105\(b\)
\(\mathrm { P } ( X = x )\)0.30.250.10.150.2
where \(b\) is a constant and \(b > 5\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(b\) Given that \(\operatorname { Var } ( X ) = 34.26\)
  2. find the value of \(b\)
  3. Find \(\mathrm { P } \left( X ^ { 2 } < 2 - 3 X \right)\)
AQA Further Paper 3 Statistics 2022 June Q6
2 marks
6 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0
b & x = 1
c & x = 2
0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The mean of \(X\) is 1.2 and the variance of \(X\) is 0.56
6
  1. Deduce the values of \(a , b\) and \(c\)
    6
  2. The continuous random variable \(Y\) is independent of \(X\) and has variance 15 Find \(\operatorname { Var } ( X - 2 Y - 11 )\)
    [0pt] [2 marks]