Probability & Statistics 2 — Variant 2

A random variable $X$ has the distribution $\mathrm{Po}(145)$.

Use a suitable approximating distribution to calculate $\mathrm{P}(X \leqslant 150)$.

Justify the use of your approximating distribution in this case.

A student wishes to estimate the proportion, $p$, of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate $\alpha\%$ confidence interval for $p$ and finds that the lower limit of the confidence interval is $0.244$ correct to 3 significant figures. Find $\alpha$ correct to the nearest integer.

A random variable $X$ has the distribution $\mathrm{N}(10, 12)$. Two independent values of $X$, denoted by $X_{1}$ and $X_{2}$, are chosen at random.

Write down the value of $\mathrm{P}(X_{1} > X_{2})$.

Find $\mathrm{P}(X_{1} > 2X_{2} - 3)$.

The number of goals scored by a sports team in the first half of any match has the distribution $X \sim \mathrm{Po}(3.1)$. The number of goals scored by the same team in the second half of any match has the distribution $Y \sim \mathrm{Po}(2.4)$. You may assume that the distributions of $X$ and $Y$ are independent.

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

Find the probability that, in a randomly chosen match, the team scores at least 5 goals.

Given that the team scores a total of 5 goals in a randomly chosen match, find the probability that they score exactly 3 goals in the first half.