Mean and Variance of Linear Combination of Variables

A hot drink stand sells coffee and chocolates on average they sell one coffee every $5$min and one hot chocolate every $15$min.Then the probability that they sell at least one hot drink in $10$mins is _____.

Let $X_1$ be a normal random variable with mean $2$ and variance $3$, and let $X_2$ be a normal random variable with mean $1$ and variance $4$. Assume that $X_1$ and $X_2$ are independent. What is the distribution of the linear combination $Y=X_1-X_2$?

A hot drink stand sells coffee and chocolates on average they sell one coffee every $5$min and one hot chocolate every $15$min. Find the probability that they sell at least one hot drink in $10$mins.

Let $X_1$ be a normal random variable with mean $2$ and variance $3$, and let $X_2$ be a normal random variable with mean $1$ and variance $4$. Assume that $X_1$ and $X_2$ are independent. What is the distribution of the linear combination $Y=2X_1+3X_2$?

The lengths of red pencils are normally distributed with mean $6.5 \mathrm{cm}$ and standard deviation $0.23 \mathrm{cm}$.

Find the probability that the total length of $3$ red pencils is more than $6.7 \mathrm{cm}$ greater than the length of $1$black pencil. 

The lengths of red pencils are normally distributed with mean $6.5 \mathrm{cm}$ and standard deviation $0.23 \mathrm{cm}$.

Two red pencils are chosen at random. Find the probability that their total length is greater than $12.5 \mathrm{cm}$. The lengths of black pencils are normally distributed with mean $11.3 \mathrm{cm}$ and standard deviation $0.46 \mathrm{cm}$ 

In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables $X$ and $Y$ , respectively, where $X~\mathrm{N}\left(57, 13\right)$ and $Y~\mathrm{N}\left(28, 5\right)$ You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating $X+2.5Y$. A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate:

obtains at least $20$ more marks in the theory paper than in the practical paper.

In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables $X$ and $Y$, respectively, where $X~\mathrm{N}\left(57, 13\right)$ and $Y~\mathrm{N}\left(28, 5\right)$. You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating $X+2.5Y$. A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate:

has final score that is greater than $140$. 

The maximum load an elevator can carry is $600 \mathrm{kg}$. The masses of men are normally distributed with mean $80 \mathrm{kg}$ and standard deviation $9 \mathrm{kg}$. The masses of women are normally distributed with mean $65 \mathrm{kg}$ and standard deviation $6 \mathrm{kg}$. Assuming the masses of men and women are independent, find the probability that the elevator will not be overloaded by a group of six men and two women.

Random variables $R$ and $S$ are such that $R~\mathrm{N}\left(\mu , 2^2\right)$ and $S~\mathrm{N}\left(2\mu , 3^2\right)$. It is given that $\mathrm{P}\left(R+S>1\right)=0.9$.

Hence, find $\mathrm{P}\left(S>R\right)$.

Random variables $R$ and $S$are such that $R~\mathrm{N}\left(\mu , 2^2\right)$ and $S~\mathrm{N}\left(2\mu , 3^2\right)$. It is given that $\mathrm{P}\left(R+S>1\right)=0.9$

Find $\mu$.

Random variables $Y$ and $X$ are related by $Y=a+bX$, where $a$ and $b$ are constants and $b>0$. The standard deviation of $Y$ is twice the standard deviation of $X$. The mean of $Y$ is $7.92$ and is $0.8$ more than mean of $X$. Find the values of $a$ and $b$.

The weekly distance, in kilometres, driven by Mr Parry has a normal distribution with mean $512$ and standard deviation $62$ Independently, the weekly distance, in kilometres, driven by Mrs Parry has a normal distribution with mean $89$ and standard deviation $7.4$.

Find the mean and standard deviation of the total of the weekly distance, in miles, driven by Mr Parry and Mrs Parry. Use the approximation $8$ kilometres$=5$ miles.

The weekly distance, in kilometres, driven by Mr Parry has a normal distribution with mean $512$ and standard deviation $62$. Independently, the weekly distance, in kilometres, driven by Mrs Parry has a normal distribution with mean $89$ and standard deviation $7.4$.

Find the probability that, in a randomly chosen week, Mr Parry drives more than five times as far as Mrs Parry.  

The masses, in milligrams, of three minerals found in $1$ tonne of a certain kind of rock are modelled by three independent random variables $P, Q$ and $R$, where $P~\mathrm{N}\left(46, {19}^2\right), Q~\mathrm{N}\left(53, {23}^2\right)$ and $R~\mathrm{N}\left(25, {10}^2\right)$. The total value of the minerals found in $1$ tonne of rock is modelled by the random variable $V$, where $V=P+Q+2R$. Use the model to find the probability of finding minerals with a value of at least $93$ in a randomly chosen tonne of rock.

The cost of hiring a bicycle consists of fixed charge of $500$ cents together with a charge of $3$ cents per minute. The number of minutes for which people hire bicycle has mean $142$ and standard deviation $35$.

Six people hire bicycle independently. Find the mean and standard deviation of the total amount paid by all six people.

The cost of hiring a bicycle consists of fixed charge of $500$ cents together with a charge of $3$ cents per minute. The number of minutes for which people hire bicycle has mean $142$ and standard deviation $35$.

Find the mean and standard deviation of the amount people pay when hiring a bicycle. 

The mean and variance of the random variable $X$ are $5.8$ and $3.1$, respectively. The random variable $S$ is the sum of three independent values of $X$. The independent random variable $T$ is defined by $T=3X+2$.

Find the mean and variance of $S-T$.

The mean and variance of the random variable $X$ are $5.8$ and $3.1$, respectively. The random variable $S$ is the sum of three independent values of $X$. The independent random variable $T$ is defined by $T=3X+2$.

Find the variance of $T$.

The mean and variance of the random variable $X$ are $5.8$ and $3.1$, respectively. The random variable $S$ is the sum of three independent values of $X$. The independent random variable $T$ is defined by $T=3X+2$.

Find the variance of $S$.