DP Mathematics HL Questionbank

• Syllabus

7.1

Sub sections and their related questions

Cumulative distribution functions for both discrete and continuous distributions.

• 12M.3sp.hl.TZ0.4d: (i)     Find the cumulative distribution function for X . (ii)     Hence, or otherwise, find the...
• SPNone.2.hl.TZ0.3a: Complete the grouped frequency table for these data.  
• SPNone.2.hl.TZ0.3b: Estimate the mean and standard deviation of the heights of these 80 boys.
• SPNone.2.hl.TZ0.3c: Explain briefly whether or not the normal distribution provides a suitable model for this...
• 13M.3sp.hl.TZ0.4a: (i)     Determine an expression for $F(x)$, valid for $1 \leqslant x \leqslant 2$, where F...
• 11N.3sp.hl.TZ0.3b: Find the cumulative distribution function, $F(x)$, of X.
• 11N.3sp.hl.TZ0.3c: Find the probability that the lifetime of a particular battery is more than twice the mean.
• 11N.3sp.hl.TZ0.3d: Find the median of X in terms of $\lambda$.
• 11N.3sp.hl.TZ0.3e: Find the probability that the lifetime of a particular battery lies between the median and the mean.
• 11M.2.hl.TZ1.1a: Estimate the (i)     median weight of the apples; (ii)     30th percentile of the weight of the...
• 11M.2.hl.TZ1.1b: Estimate the number of apples which weigh more than 110 grams.
• 14N.3sp.hl.TZ0.1b: Find the cumulative distribution function for $X$.
• 14N.3sp.hl.TZ0.1c: Find the interquartile range for $X$.
• 15N.3sp.hl.TZ0.4a: Find $F(u)$, the cumulative distribution function of $U$, for...
• 15N.3sp.hl.TZ0.4b: Hence, or otherwise, find the value of $P(U > 20)$.
• 16M.3sp.hl.TZ0.5a: Show that the probability distribution of $Y$ is given by...
• 16M.3sp.hl.TZ0.5b: (i)     Show that $G(t)$, the probability generating function of $Y$, is given by...
• 16N.3sp.hl.TZ0.4a: Write down an expression for $J(t)$ in terms of $G(t)$ and $H(t)$.
• 16N.3sp.hl.TZ0.4b: By differentiating $J(t)$, prove that (i)    ...
• 17N.3sp.hl.TZ0.1a: Find the cumulative distribution function $F(t)$, for $0 \leqslant t \leqslant 2$.
• 17N.3sp.hl.TZ0.1b.i: Sketch the graph of $F(t)$ for $0 \leqslant t \leqslant 2$, clearly indicating the...
• 17N.3sp.hl.TZ0.1b.ii: Given that $P(T < a) = 0.75$, find the value of $a$.
• 17N.3sp.hl.TZ0.5a.i: Find expressions for ${G’_X}(t)$ and ${G’’_X}(t)$.
• 17N.3sp.hl.TZ0.5a.ii: Hence show that ${\text{Var}}(X) = \lambda$.
• 17N.3sp.hl.TZ0.5b: By considering the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, show that...
• 17N.3sp.hl.TZ0.5c.i: Show...
• 17N.3sp.hl.TZ0.5c.ii: Identify the probability distribution given in part (c)(i) and state its parameters.
• 18M.3sp.hl.TZ0.2a: State the distribution of X.
• 18M.3sp.hl.TZ0.2b: Show that the probability generating function, $G\left( t \right)$, for X is given by...
• 18M.3sp.hl.TZ0.2c: Find $G'\left( t \right)$.
• 18M.3sp.hl.TZ0.2d: Determine the mean number of throws required to obtain a 1.

Geometric distribution.

• 12M.3sp.hl.TZ0.2a: Show that ${\text{P}}(X \leqslant n) = 1 - {(1 - p)^n},{\text{ }}n \in {\mathbb{Z}^ + }$ .
• 12M.3sp.hl.TZ0.2b: Deduce an expression for...
• 12M.3sp.hl.TZ0.2c: Given that p = 0.2, find the least value of n for which...
• 12N.3sp.hl.TZ0.4a: If the dice is fair, write down the distribution of X , including the value of any parameter(s).
• 12N.3sp.hl.TZ0.4b: Write down E(X ) for the distribution in part (a).
• 08M.3sp.hl.TZ1.5: (a)     Having thrown the die once, she lets ${X_2}$ denote the number of additional throws...
• 09M.3sp.hl.TZ0.4: In a game there are n players, where $n > 2$ . Each player has a disc, one side of which is...
• 14M.3sp.hl.TZ0.4: Consider the random variable $X \sim {\text{Geo}}(p)$. (a)     State \({\text{P}}(X <...
• 15N.3sp.hl.TZ0.4a: Find $F(u)$, the cumulative distribution function of $U$, for...
• 16M.3sp.hl.TZ0.5a: Show that the probability distribution of $Y$ is given by...
• 16M.3sp.hl.TZ0.5b: (i)     Show that $G(t)$, the probability generating function of $Y$, is given by...
• 16N.3sp.hl.TZ0.4a: Write down an expression for $J(t)$ in terms of $G(t)$ and $H(t)$.
• 16N.3sp.hl.TZ0.4b: By differentiating $J(t)$, prove that (i)    ...
• 17N.3sp.hl.TZ0.1a: Find the cumulative distribution function $F(t)$, for $0 \leqslant t \leqslant 2$.
• 17N.3sp.hl.TZ0.1b.i: Sketch the graph of $F(t)$ for $0 \leqslant t \leqslant 2$, clearly indicating the...
• 17N.3sp.hl.TZ0.1b.ii: Given that $P(T < a) = 0.75$, find the value of $a$.
• 17N.3sp.hl.TZ0.5a.i: Find expressions for ${G’_X}(t)$ and ${G’’_X}(t)$.
• 17N.3sp.hl.TZ0.5a.ii: Hence show that ${\text{Var}}(X) = \lambda$.
• 17N.3sp.hl.TZ0.5b: By considering the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, show that...
• 17N.3sp.hl.TZ0.5c.i: Show...
• 17N.3sp.hl.TZ0.5c.ii: Identify the probability distribution given in part (c)(i) and state its parameters.
• 18M.3sp.hl.TZ0.2a: State the distribution of X.
• 18M.3sp.hl.TZ0.2b: Show that the probability generating function, $G\left( t \right)$, for X is given by...
• 18M.3sp.hl.TZ0.2c: Find $G'\left( t \right)$.
• 18M.3sp.hl.TZ0.2d: Determine the mean number of throws required to obtain a 1.

Negative binomial distribution.

• 12N.3sp.hl.TZ0.4d: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice....
• 12N.3sp.hl.TZ0.4e: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice....
• 12N.3sp.hl.TZ0.4f: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice....
• 08N.3sp.hl.TZ0.3: (a)     The heating in a residential school is to be increased on the third frosty day during the...
• 11M.3sp.hl.TZ0.5a: The random variable X has the negative binomial distribution NB(3, p) . Let $f(x)$ denote the...
• 11M.3sp.hl.TZ0.5b: Jo has a biased coin which has a probability of 0.35 of showing heads when tossed. She tosses...
• 10M.3sp.hl.TZ0.5b: The random variable X has the negative binomial distribution NB(5, p), where p < 0.5, and...
• 10N.3sp.hl.TZ0.3: As soon as Sarah misses a total of 4 lessons at her school an email is sent to her parents. The...
• 13M.3sp.hl.TZ0.5a: Find the probability that (i)     he hits the target exactly 4 times in his first 8 shots; (ii)...
• 13M.3sp.hl.TZ0.5b: Ben hits the target for the ${10^{{\text{th}}}}$ time with his ${X^{{\text{th}}}}$ shot. (i)...
• 14N.3sp.hl.TZ0.2b: (i)     Find the expected number of throws required for Eric to hit the target three times. (ii)...
• 14N.3sp.hl.TZ0.2c: If he has just $$8$, find the probability he will lose all his money before he hits the target...
• 16M.3sp.hl.TZ0.5a: Show that the probability distribution of $Y$ is given by...
• 16M.3sp.hl.TZ0.5b: (i)     Show that $G(t)$, the probability generating function of $Y$, is given by...
• 16N.3sp.hl.TZ0.4a: Write down an expression for $J(t)$ in terms of $G(t)$ and $H(t)$.
• 16N.3sp.hl.TZ0.4b: By differentiating $J(t)$, prove that (i)    ...
• 17N.3sp.hl.TZ0.1a: Find the cumulative distribution function $F(t)$, for $0 \leqslant t \leqslant 2$.
• 17N.3sp.hl.TZ0.1b.i: Sketch the graph of $F(t)$ for $0 \leqslant t \leqslant 2$, clearly indicating the...
• 17N.3sp.hl.TZ0.1b.ii: Given that $P(T < a) = 0.75$, find the value of $a$.
• 17N.3sp.hl.TZ0.5a.i: Find expressions for ${G’_X}(t)$ and ${G’’_X}(t)$.
• 17N.3sp.hl.TZ0.5a.ii: Hence show that ${\text{Var}}(X) = \lambda$.
• 17N.3sp.hl.TZ0.5b: By considering the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, show that...
• 17N.3sp.hl.TZ0.5c.i: Show...
• 17N.3sp.hl.TZ0.5c.ii: Identify the probability distribution given in part (c)(i) and state its parameters.
• 18M.3sp.hl.TZ0.2a: State the distribution of X.
• 18M.3sp.hl.TZ0.2b: Show that the probability generating function, $G\left( t \right)$, for X is given by...
• 18M.3sp.hl.TZ0.2c: Find $G'\left( t \right)$.
• 18M.3sp.hl.TZ0.2d: Determine the mean number of throws required to obtain a 1.

Probability generating functions for discrete random variables.

• SPNone.3sp.hl.TZ0.2b: Show that the probability generating function for X is given...
• SPNone.3sp.hl.TZ0.2c: Hence determine ${\text{E}}(X)$.
• 14N.3sp.hl.TZ0.4a: (i)     Prove that ${\text{E}}(X) = \lambda$. (ii)     Prove that...
• 14N.3sp.hl.TZ0.4e: By consideration of the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, prove...
• 14N.3sp.hl.TZ0.4g: Hence find the probability that $X + Y$ is an even number.
• 15M.3sp.hl.TZ0.5a: Determine the probability generating function for $X \sim {\text{B}}(1,{\text{ }}p)$.
• 15M.3sp.hl.TZ0.5b: Explain why the probability generating function for ${\text{B}}(n,{\text{ }}p)$ is a polynomial...
• 15M.3sp.hl.TZ0.5c: Two independent random variables ${X_1}$ and ${X_2}$ are such that...
• 15N.3sp.hl.TZ0.4c: Prove that the probability generating function of $U$ is given by...
• 16M.3sp.hl.TZ0.5a: Show that the probability distribution of $Y$ is given by...
• 16M.3sp.hl.TZ0.5b: (i)     Show that $G(t)$, the probability generating function of $Y$, is given by...
• 16N.3sp.hl.TZ0.4a: Write down an expression for $J(t)$ in terms of $G(t)$ and $H(t)$.
• 16N.3sp.hl.TZ0.4b: By differentiating $J(t)$, prove that (i)    ...
• 17N.3sp.hl.TZ0.1a: Find the cumulative distribution function $F(t)$, for $0 \leqslant t \leqslant 2$.
• 17N.3sp.hl.TZ0.1b.i: Sketch the graph of $F(t)$ for $0 \leqslant t \leqslant 2$, clearly indicating the...
• 17N.3sp.hl.TZ0.1b.ii: Given that $P(T < a) = 0.75$, find the value of $a$.
• 17N.3sp.hl.TZ0.5a.i: Find expressions for ${G’_X}(t)$ and ${G’’_X}(t)$.
• 17N.3sp.hl.TZ0.5a.ii: Hence show that ${\text{Var}}(X) = \lambda$.
• 17N.3sp.hl.TZ0.5b: By considering the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, show that...
• 17N.3sp.hl.TZ0.5c.i: Show...
• 17N.3sp.hl.TZ0.5c.ii: Identify the probability distribution given in part (c)(i) and state its parameters.
• 18M.3sp.hl.TZ0.2a: State the distribution of X.
• 18M.3sp.hl.TZ0.2b: Show that the probability generating function, $G\left( t \right)$, for X is given by...
• 18M.3sp.hl.TZ0.2c: Find $G'\left( t \right)$.
• 18M.3sp.hl.TZ0.2d: Determine the mean number of throws required to obtain a 1.

Using probability generating functions to find mean, variance and the distribution of the sum of $n$ independent random variables.

• 14N.3sp.hl.TZ0.4a: (i)     Prove that ${\text{E}}(X) = \lambda$. (ii)     Prove that...
• 14N.3sp.hl.TZ0.4e: By consideration of the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, prove...
• 14N.3sp.hl.TZ0.4f: Find (i)     ${G_{X + Y}}(1)$; (ii)     ${G_{X + Y}}( - 1)$.
• 15N.3sp.hl.TZ0.4e: A third random variable $W$, has probability generating function...
• 15N.3sp.hl.TZ0.4f: A third random variable $W$, has probability generating function...
• 16M.3sp.hl.TZ0.5a: Show that the probability distribution of $Y$ is given by...
• 16M.3sp.hl.TZ0.5b: (i)     Show that $G(t)$, the probability generating function of $Y$, is given by...
• 17N.3sp.hl.TZ0.1a: Find the cumulative distribution function $F(t)$, for $0 \leqslant t \leqslant 2$.
• 17N.3sp.hl.TZ0.1b.i: Sketch the graph of $F(t)$ for $0 \leqslant t \leqslant 2$, clearly indicating the...
• 17N.3sp.hl.TZ0.1b.ii: Given that $P(T < a) = 0.75$, find the value of $a$.
• 17N.3sp.hl.TZ0.5a.i: Find expressions for ${G’_X}(t)$ and ${G’’_X}(t)$.
• 17N.3sp.hl.TZ0.5a.ii: Hence show that ${\text{Var}}(X) = \lambda$.
• 17N.3sp.hl.TZ0.5b: By considering the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, show that...
• 17N.3sp.hl.TZ0.5c.i: Show...
• 17N.3sp.hl.TZ0.5c.ii: Identify the probability distribution given in part (c)(i) and state its parameters.
• 18M.3sp.hl.TZ0.2a: State the distribution of X.
• 18M.3sp.hl.TZ0.2b: Show that the probability generating function, $G\left( t \right)$, for X is given by...
• 18M.3sp.hl.TZ0.2c: Find $G'\left( t \right)$.
• 18M.3sp.hl.TZ0.2d: Determine the mean number of throws required to obtain a 1.