A continuous random variable $X$ has a probability density function given by

$f\left(x\right)=\left\{\begin{array}{cc}\text{arccos} x & 0\le x\le 1\\ 0& \text{otherwise}\end{array}\right.$

The median of this distribution is $m$.

A random variable $X$ has probability density function

Consider the case where $b=5$.

Find the value of

A continuous random variable $X$ has the probability density function $f_n$ given by

$f_n\left(x\right)=\left\{\begin{array}{l}\left(n+1\right)x^n,\\ 0,\end{array} \right.\begin{array}{c}0\le x\le 1\\ \text{otherwise}\end{array}$

where $n\in \mathrm{ℝ}, n\ge 0$.

The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.

It is found that 5% of the male runners complete the marathon in less than $T_1$ minutes.

The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.

A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let $\text{P}(X=n)$ be the probability that Kati obtains her third voucher on the $n\text{th}$ bar opened.

(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)

It is given that $\text{P}(X=n)=\frac{n^2+an+b}{2000}\times <!--\; \times \; -->{0.9}^{n-<!--\; -\; -->3}$ for $n⩾<!--\; ⩾\; -->3,n\in <!--\; \in \; -->\mathbb{N}$.

Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.

Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.

A random variable $X$ is normally distributed with mean $\mathrm{\mu <!--\; \mu \; -->}$ and standard deviation $\mathrm{\sigma <!--\; \sigma \; -->}$, such that and $\text{P}(X>42.52)=0.3060$.

A café serves sandwiches and cakes. Each customer will choose one of the following three options; buy only a sandwich, buy only a cake or buy both a sandwich and a cake.

The probability that a customer buys a sandwich is 0.72 and the probability that a customer buys a cake is 0.45.

Find the probability that a customer chosen at random will buy

On a typical day 200 customers come to the café.

It is known that 46 % of the customers who come to the café are male, and that 80 % of these buy a sandwich.

Packets of biscuits are produced by a machine. The weights $X$, in grams, of packets of biscuits can be modelled by a normal distribution where $X\sim <!--\; \sim \; -->\text{N}(\mathrm{\mu <!--\; \mu \; -->},{\mathrm{\sigma <!--\; \sigma \; -->}}^2)$. A packet of biscuits is considered to be underweight if it weighs less than 250 grams.

The manufacturer makes the decision that the probability that a packet is underweight should be 0.002. To do this $\mathrm{\mu <!--\; \mu \; -->}$ is increased and $\mathrm{\sigma <!--\; \sigma \; -->}$ remains unchanged.

The manufacturer is happy with the decision that the probability that a packet is underweight should be 0.002, but is unhappy with the way in which this was achieved. The machine is now adjusted to reduce $\mathrm{\sigma <!--\; \sigma \; -->}$ and return $\mathrm{\mu <!--\; \mu \; -->}$ to 253.

A continuous random variable $X$ has the probability density function $f$ given by

$f\left(x\right)=\left\{\begin{array}{l}\frac{x}{\sqrt{{\left(x^2+k\right)}^3}} 0\le x\le 4\\ 0 \text{otherwise}\end{array}\right.$

where $k\in {\mathrm{ℝ}}^{+}$.

The following table below shows the marks scored by seven students on two different mathematics tests.

Let L₁ be the regression line of x on y. The equation of the line L₁ can be written in the form x = ay + b.

Let L₂ be the regression line of y on x. The lines L₁ and L₂ pass through the same point with coordinates (p , q).

In Lucy’s music academy, eight students took their piano diploma examination and achieved scores out of 150. For her records, Lucy decided to record the average number of hours per week each student reported practising in the weeks prior to their examination. These results are summarized in the table below.

In a city, the number of passengers, X, who ride in a taxi has the following probability distribution.

After the opening of a new highway that charges a toll, a taxi company introduces a charge for passengers who use the highway. The charge is $ 2.40 per taxi plus $ 1.20 per passenger. Let T represent the amount, in dollars, that is charged by the taxi company per ride.

The age, L, in years, of a wolf can be modelled by the normal distribution L ~ N(8, 5).

The random variable X has a binomial distribution with parameters n and p.
It is given that E(X) = 3.5.

The continuous random variable X has probability density function $f$ given by

A continuous random variable $X$ has probability density function $f$ given by

It is given that $\text{P}(X⩾<!--\; ⩾\; -->2)=0.75$.

Eight independent observations of $X$ are now taken and the random variable $Y$ is the number of observations such that $X⩾<!--\; ⩾\; -->2$.

The random variable X has a normal distribution with mean μ = 50 and variance σ ² = 16 .

Timmy owns a shop. His daily income from selling his goods can be modelled as a normal distribution, with a mean daily income of $820, and a standard deviation of $230. To make a profit, Timmy’s daily income needs to be greater than $1000.

Consider two events $A$ and $B$ such that $\text{P}(A)=k,\text{ P}(B)=3k,\text{ P}(A\cap <!--\; \cap \; -->B)=k^2$ and $\text{P}(A\cup <!--\; \cup \; -->B)=0.5$.

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{g}$ and standard deviation $8 \text{g}$.

A random variable $X$ has a probability distribution given in the following table.

The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

The marks achieved by eight students in a class test are given in the following list.

The teacher increases all the marks by 2. Write down the new value for

Events $A$ and $B$ are such that $\text{P}(A\cup <!--\; \cup \; -->B)=0.95,\text{ P}(A\cap <!--\; \cap \; -->B)=0.6$ and $\text{P}(A|B)=0.75$.

It is known that 56 % of Infiglow batteries have a life of less than 16 hours, and 94 % have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution $\text{N}\left(\mathrm{\mu <!--\; \mu \; -->},{\mathrm{\sigma <!--\; \sigma \; -->}}^2\right)$.

Steffi the stray cat often visits Will’s house in search of food. Let $X$ be the discrete random variable “the number of times per day that Steffi visits Will’s house”.

The random variable $X$ can be modelled by a Poisson distribution with mean 2.1.

Let Y be the discrete random variable “the number of times per day that Steffi is fed at Will’s house”. Steffi is only fed on the first four occasions that she visits each day.

A discrete random variable $X$ follows a Poisson distribution $\text{Po}(\mathrm{\mu <!--\; \mu \; -->})$.