Probability & Statistics 2 — Variant 2

A random variable $X$ has the distribution $\mathrm{Po}(25)$. Use the normal approximation to the Poisson distribution to find $\mathrm{P}(X > 30)$.

A random variable $Y$ has the distribution $\mathrm{B}(100, p)$ where $p < 0.05$. Use the Poisson approximation to the binomial distribution to write down an expression, in terms of $p$, for $\mathrm{P}(Y < 3)$.

The random variable $X$ has probability density function, $f$, given by $f(x) = \begin{cases} \frac{1}{x^{2}} & a < x < b, \\ 0 & \text{otherwise,} \end{cases}$ where $a$ and $b$ are positive constants.

It is given that $\mathrm{E}(X) = \ln 2$. Show that $b = 2a$.

Show that $a = \frac{1}{2}$.

Find the median of $X$.

A factory makes loaves of bread in batches. One batch of loaves contains $X$ kilograms of dried yeast and $Y$ kilograms of flour, where $X$ and $Y$ have the independent distributions $\mathrm{N}(0.7, 0.02^{2})$ and $\mathrm{N}(100.0, 3.0^{2})$ respectively. Dried yeast costs $\$13.50$ per kilogram and flour costs $\$0.90$ per kilogram. For making one batch of bread the total of all other costs is $\$55$. The factory sells each batch of bread for $\$200$. Find the probability that the profit made on one randomly chosen batch of bread is greater than $\$40$.

A random variable $X$ has the distribution $\mathrm{Po}(2.4)$.

Find $\mathrm{P}(2 \leqslant X < 4)$.

Two independent values of $X$ are chosen. Find the probability that both of these values are greater than $1$.

It is given that $\mathrm{P}(X = r) < \mathrm{P}(X = r + 1)$.

Find the set of possible values of $r$.

Hence find the value of $r$ for which $\mathrm{P}(X = r)$ is greatest.