The number of customers who purchase a warranty during one day is modelled by $X\sim B(24,p)$. It is known that $E(X)=9.6$.

The time $X$, in minutes, taken by a runner to complete a route is normally distributed. The $20$th percentile is $42$ minutes and the $80$th percentile is $58$ minutes.

The random variables $X$ and $Y$ represent the numbers of two types of item in a shipment. They are independent, with $E(X)=18$, $\operatorname{Var}(X)=12$, $E(Y)=10$ and $\operatorname{Var}(Y)=6$. The shipment cost is modelled by $C=4X+7Y+50$.

The numbers of faults detected in one hour by two independent monitoring systems are represented by $X\sim\operatorname{Po}(2.8)$ and $Y\sim\operatorname{Po}(1.6)$.

A communications company studies two situations. In situation A, each of $50$ independently transmitted messages has probability $0.04$ of containing an error. In situation B, independent network outages occur at a uniform mean rate of $2.1$ per week.

The number of customers arriving at a kiosk in one hour is modelled by $X\sim\operatorname{Po}(\lambda)$. Assume that customers arrive according to a homogeneous Poisson process with a constant rate. The probability that no customers arrive in one hour is $0.05$.

A quality score is defined by $Y=100-5X$, where $X\sim B(20,0.30)$.

At a school fair, the number of tokens won in a game is represented by the discrete random variable $X$. Its probability distribution is shown in the table.

A wildlife survey estimates that each photograph taken by a motion-sensitive camera has probability $0.28$ of showing a fox. The photographs are assumed to be independent. Let $X$ be the number showing a fox among the next $20$ photographs.

The operating time $T$, in hours, of a rechargeable lamp is normally distributed with mean $42$ hours and standard deviation $5.5$ hours.

An insurer offers one-year protection for a portable music player. The claim payment $C$, in dollars, for one policy has the distribution shown in the table.

A commuter uses a bicycle-sharing station on eight mornings. On each morning, the probability that no bicycle is available is $p$. Availability on different mornings is independent. The probability that a bicycle is available on all eight mornings is $0.12$.

The daily energy use $X$, in kilowatt-hours, of a particular household is normally distributed. It is known that $P(X<120)=0.10$ and $P(X<168)=0.90$.

A basketball player makes a free throw with probability $0.60$. The outcomes of successive free throws are independent. During a training exercise, the player takes $15$ free throws. Let $X$ be the number made.

The lifetime $L$, in days, of a water-purification cartridge is normally distributed with mean $860$ days and standard deviation $45$ days.

The number of maintenance tasks completed by a technician during one session is represented by the random variable $X$ with the distribution shown in the table.

Two independent random variables $X$ and $Y$ represent the masses, in kilograms, of two materials used in a manufacturing process. It is known that $E(X)=2.4$, $\operatorname{Var}(X)=0.09$, $E(Y)=1.7$ and $\operatorname{Var}(Y)=0.04$. The production cost, in dollars, is modelled by $C=12X+8Y+5$, and a balance index is defined by $D=3X-2Y$.

A random sample of travel times, in minutes, is summarized in the frequency table.

Small cracks along a walking trail occur independently at a uniform mean rate of $4.8$ cracks per $3$ kilometres. Let $X$ be the number of cracks along a randomly selected $5$ kilometre section.

At a school fair, a game produces a score $X$ with the probability distribution shown. The player receives $6X$ credits. Different plays are independent.

A laboratory independently tests $30$ water samples. Each sample has probability $0.08$ of containing a pollutant above a specified limit. Let $X$ be the number of samples above the limit.

The time $X$, in minutes, taken by an automated device to complete a cycle is normally distributed. Records show that $P(X<18.4)=0.10$ and $P(X>21.6)=0.10$.

A delivery company records the daily numbers $X$ and $Y$ of two types of special delivery. The variables are independent, with $E(X)=2.5$, $\operatorname{Var}(X)=2.5$, $E(Y)=3$ and $\operatorname{Var}(Y)=2.55$. Daily operating cost is modelled by $C=120+8X+3Y$.

A random sample records the number of maintenance tasks completed by workers during one shift.

An engineering team studies three processes: errors among $60$ independent transmissions, each with probability $0.03$ of error; cracks occurring independently along cable at a uniform mean rate of $0.4$ per metre; and the mass of a component, which is continuous and approximately symmetric with mean $10$ grams and standard deviation $0.3$ grams.

For each policy sold, an insurer pays a claim $X$ of $0$, $500$ or $2000$ monetary units with probabilities $0.92$, $0.07$ and $0.01$, respectively. The insurer also has an administration cost of $12$ monetary units per policy.

A temperature sensor reports a reading $X$ in arbitrary units. The readings are normally distributed with mean $52$ and standard deviation $4$. The calibrated temperature is defined by $Y=aX+b$. Two calibration requirements are $E(Y)=20$ and $\operatorname{sd}(Y)=3$.

A random sample of five soil-moisture readings, expressed as percentages, is $18$, $21$, $23$, $24$ and $29$. Two estimators of population variance are considered: $V_1=s_n^2$ and $V_2=s_{n-1}^2$.