The first two moments of the distribution about the value $5$ of the variable, are $2$ and $20$. Find the mean and the variance.

When a guitar is played regularly, a string breaks on average once every $15$ months. Broken strings occur at random times and independently of each other.

The mean number of broken strings of the new type in a $5$-year period is denoted by $\lambda$. Find the rejection region for a test at the $10\%$ significanc e level when the null hypothesis $\lambda =4$ is tested against the alternative hypothesis $\lambda <4$.

The independent random variable $X$ and $Y$ have the distribution $\mathrm{Po}\left(2\right)$ and $\mathrm{Po}\left(3\right)$, respectively.

Given that $X+Y=5$, find the probability that $X=1$ and $Y=4$.

The number of lions seen per day during a standard safari has the distribution $\mathrm{Po}\left(0.8\right)$. The number of lions seen per day during an off-road safari has the distribution $\mathrm{Po}\left(2.7\right)$. The two distributions are independent.

Deena goes on a standard safari for $3$ days and then on an off-road safari for $2$ days. Find the probability that she sees a total of fewer than $5$ lions.

The independent random variable $X$ and $Y$ have the distribution $\mathrm{Po}\left(2\right)$ and $\mathrm{Po}\left(3\right)$, respectively.

Given that $\mathrm{P}\left(X=r\right)=\frac{2}{3}\mathrm{P}\left(X=0\right)$, show that $3\times 2^{r-1}=r!$

The number of accidents on a certain road has a Poisson distribution with mean $3.1$ per $12$-week peroid 

Given that the mean number of accidents per week is now $0.1$, find the probability of a Type $\mathrm{II}$ error.

When a guitar is played regularly, a string breaks on average once every $15$ months. Broken strings occur at random times and independently of each other.

Show that the mean number of broken strings in a $5$-year period is $4$.

A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of $5$ years.

Represent the union of two sets by Venn diagram for each of the following.

$X=\{x | x$ is a prime number between $80$ and $100\}$

$Y=\{y | y$ is an odd number between $90$ and $100\}$

A random variable $X$ has the distribution $\mathrm{Po}\left(1.6\right)$

The random variable $R$ is the sum of three independent values of $X$. Find $\mathrm{P}\left(R<4\right)$.

The number of lions seen per day during a standard safari has the distribution $\mathrm{Po}\left(0.8\right)$. The number of lions seen per day during an off-road safari has the distribution $\mathrm{Po}\left(2.7\right)$. The two distributions are independent.

Susan goes on a standard safari for one day. Find the probability that she sees at least $2$ lions.

A random variable $X$ has the distribution $\mathrm{Po}\left(1.6\right)$

The random variable $S$ is the sum of $n$ independent values of $X$. It is given that $\mathrm{P}\left(S=4\right)=\frac{16}{3}\times \mathrm{P}\left(S=2\right)$. Find $n$.

The number of accidents on a certain road has a Poisson distribution with mean $3.1$ per $12$-week period.

Find the probability of a Type $\mathrm{I}$ error.

The number of lions seen per day during a standard safari has the distribution $\mathrm{Po}\left(0.8\right)$. The number of lions seen per day during an off-road safari has the distribution $\mathrm{Po}\left(2.7\right)$. The two distributions are independent.

The number of accidents on a certain road has a Poisson distribution with mean $3.1$ per $12$-week period.

Given that there were $2$ accidents during the $15$ weeks, explain why it is impossible for the officer to make a Type $\mathrm{II}$ error.

A random variable $X$ has the distribution $\mathrm{Po}\left(1.6\right)$

The random variable$T$ is the sum of $40$ independent values of $X$. Find $\mathrm{P}\left(T>75\right)$