Simple algebraic expression for P(X=x)

Probabilities given as a single algebraic expression in x (e.g., kx, kx², k(x²-1), k(x+1)) where k is found by summing over all values of x and setting equal to 1.

32 questions · Moderate -0.7

5.02b Expectation and variance: discrete random variables
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OCR MEI Further Statistics Minor 2021 November Q1
7 marks Moderate -0.8
1 The probability distribution of a discrete random variable \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } \left( ( \mathrm { r } - 1 ) ^ { 2 } + 1 \right)\) for \(r = 1,2,3,4,5\).
  1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.
    \(r\)12345
    \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)\(\frac { 1 } { 35 }\)\(\frac { 2 } { 35 }\)\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 17 } { 35 }\)
  2. Comment briefly on the shape of the distribution.
  3. Find each of the following.
    The random variable \(Y\) is given by \(Y = 5 X - 10\).
  4. Find each of the following.
OCR MEI Further Statistics Major 2019 June Q1
11 marks Moderate -0.3
1 A fair six-sided dice is rolled three times.
The random variable \(X\) represents the lowest of the three scores.
The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).
  1. Complete the copy of the table in the Printed Answer Booklet.
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(91 k\)\(61 k\)\(37 k\)
  2. Show that \(k = \frac { 1 } { 216 }\).
  3. Draw a graph to illustrate the distribution.
  4. Comment briefly on the shape of the distribution.
  5. In this question you must show detailed reasoning. Find each of the following.
AQA AS Paper 2 2019 June Q14
4 marks Easy -1.2
14 A probability distribution is given by $$\mathrm { P } ( X = x ) = c ( 4 - x ) , \text { for } x = 0,1,2,3$$ where \(c\) is a constant.
14
  1. Show that \(c = \frac { 1 } { 10 }\) 14
  2. Calculate \(\mathrm { P } ( X \geq 1 )\)
Edexcel S1 Q4
11 marks Moderate -0.3
The discrete random variable \(X\) has probability function P\((X = x) = k(x + 4)\). Given that \(X\) can take any of the values \(-3, -2, -1, 0, 1, 2, 3, 4\),
  1. find the value of the constant \(k\). [3 marks]
  2. Find P\((X < 0)\). [2 marks]
  3. Show that the cumulative distribution F\((x)\) is given by $$\text{F}(x) = \lambda(x + 4)(x + 5)$$ where \(\lambda\) is a constant to be found. [6 marks]
OCR S1 2013 January Q1
7 marks Moderate -0.8
When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values 2, 4, 6 and 8. The spinner is biased so that P(\(X = x\)) = \(kx\), where \(k\) is a constant.
  1. Show that P(\(X = 6\)) = \(\frac{3}{10}\). [2]
  2. Find E(\(X\)) and Var(\(X\)). [5]
OCR MEI S1 2011 January Q4
7 marks Standard +0.3
The probability distribution of the random variable \(X\) is given by the formula $$\text{P}(X = r) = kr(r + 1) \quad \text{for } r = 1, 2, 3, 4, 5.$$
  1. Show that \(k = \frac{1}{70}\). [2]
  2. Find E\((X)\) and Var\((X)\). [5]
Edexcel S1 Q5
13 marks Moderate -0.8
The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Find the value of \(k\). [2 marks]
  2. Show that E\((X) = \frac{9}{2}\). [3 marks]
Find
  1. P\([X > \text{E}(X)]\), [2 marks]
  2. E\((2X - 5)\), [2 marks]
  3. Var\((X)\). [4 marks]