DP Further Mathematics HL Questionbank
3.1
Description
[N/A]Directly related questions
- 18M.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
- 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
- 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
- 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
- 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
- 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
- 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
- 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
- 17M.2.hl.TZ0.7b.iii: Hence find \({\text{Var}}(X)\).
- 17M.2.hl.TZ0.7b.ii: By differentiating both sides of this equation, determine the values of \(G’(1)\) and \(G’’(1)\).
- 17M.2.hl.TZ0.7b.i: Show that \(\ln G(t) = \ln 4 + \ln t - 2\ln (3 - t)\).
- 17M.2.hl.TZ0.7a.iv: Hence show that \(k = \frac{4}{3}\).
- 17M.2.hl.TZ0.7a.iii: By considering \(\left( {1 - \frac{t}{3}} \right)G(t)\), show...
- 17M.2.hl.TZ0.7a.ii: Determine the radius of convergence of this infinite series.
- 17M.2.hl.TZ0.7a.i: Write down the first three terms of the infinite series for \(G(t)\), the probability generating...
- 15M.2.hl.TZ0.6c: Find \({\text{P}}(A \leqslant 5|A > 3)\).
- 15M.2.hl.TZ0.6b: Show that...
- 15M.2.hl.TZ0.6a: Find \({\text{E}}(A)\).
- 09M.2.hl.TZ0.1A.b: (i) Use your answer to (a) to find an approximate expression for the cumulative distributive...
- 13M.2.hl.TZ0.3a: (i) Find an expression for \({\rm{P}}(X > a)\) , where \(a > 0\) . A chicken crosses a...
- 13M.2.hl.TZ0.3b: A rifleman shoots at a circular target. The distance in centimetres from the centre of the target...
- 13M.2.hl.TZ0.1a: (i) Write down the mode of \(X\) . (ii) Find the exact value of \(p\) if...
- 13M.2.hl.TZ0.1b: (i) Find the smallest value of \(n\) for which the probability of Arthur walking to school on...
- 12M.2.hl.TZ0.5a: The continuous random variable \(X\) takes values only in the interval [\(a\), \(b\)] and \(F\)...
- 12M.2.hl.TZ0.5b: The continuous random variable \(Y\) has probability density function \(f\) given...
- SPNone.2.hl.TZ0.7b: (i) Show that \({\rm{P}}(Y \le y) = {\left( {\frac{y}{a}} \right)^{3n}},0 \le y \le a\) and...
- 14M.1.hl.TZ0.3: The following table shows the probability distribution of the discrete random variable...
Sub sections and their related questions
Cumulative distribution functions for both discrete and continuous distributions.
- 09M.2.hl.TZ0.1A.b: (i) Use your answer to (a) to find an approximate expression for the cumulative distributive...
- 13M.2.hl.TZ0.3a: (i) Find an expression for \({\rm{P}}(X > a)\) , where \(a > 0\) . A chicken crosses a...
- 13M.2.hl.TZ0.3b: A rifleman shoots at a circular target. The distance in centimetres from the centre of the target...
- 12M.2.hl.TZ0.5a: The continuous random variable \(X\) takes values only in the interval [\(a\), \(b\)] and \(F\)...
- 12M.2.hl.TZ0.5b: The continuous random variable \(Y\) has probability density function \(f\) given...
- SPNone.2.hl.TZ0.7b: (i) Show that \({\rm{P}}(Y \le y) = {\left( {\frac{y}{a}} \right)^{3n}},0 \le y \le a\) and...
- 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
- 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
- 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
- 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
- 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
- 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
- 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
- 18M.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
Geometric distribution.
- 13M.2.hl.TZ0.1a: (i) Write down the mode of \(X\) . (ii) Find the exact value of \(p\) if...
- 15M.2.hl.TZ0.6a: Find \({\text{E}}(A)\).
- 15M.2.hl.TZ0.6b: Show that...
- 15M.2.hl.TZ0.6c: Find \({\text{P}}(A \leqslant 5|A > 3)\).
- 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
- 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
- 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
- 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
- 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
- 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
- 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
- 18M.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
Negative binomial distribution.
- 13M.2.hl.TZ0.1b: (i) Find the smallest value of \(n\) for which the probability of Arthur walking to school on...
- 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
- 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
- 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
- 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
- 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
- 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
- 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
- 18M.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
Probability generating functions for discrete random variables.
- 14M.1.hl.TZ0.3: The following table shows the probability distribution of the discrete random variable...
- 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
- 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
- 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
- 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
- 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
- 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
- 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
- 18M.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
Using probability generating functions to find mean, variance and the distribution of the sum of n independent random variables.
- 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
- 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
- 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
- 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
- 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
- 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
- 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
- 18M.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
