AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators Questionbank

A quality control officer tests the resistance (in ohms) of resistors produced in a factory, which follows a normal distribution with mean 100 ohms and variance 16 ohms $^{2}$. A sample of 12 resistors is taken, with $\sum x=1194$ ohms and $\sum x^{2}=119108$ ohms $^{2}$.

A random variable $Z$ is normally distributed with mean $10$ and variance $4$. A linear transformation is defined as $W = 3Z_1 + 2Z_2$, where $Z_1$ and $Z_2$ are independent observations of $Z$.

The continuous random variable $X$ has probability density function $f$ given by

$$ f(x) = \begin{cases} 3ax, & 0 \leqslant x < 0.5 \\ a(2 - x), & 0.5 \leqslant x < 2 \\ 0, & \text{otherwise} \end{cases} $$

Two independent random variables $X$ and $Y$ follow Poisson distributions.

Given that $\text{E}(X) = 3$ and $\text{E}(Y) = 4$, calculate:

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, such that $P(X < 30.31) = 0.1180$ and $P(X > 42.52) = 0.3060$.

Let $W_1 \sim \text{Poisson}(5)$ and $W_2 \sim \text{Poisson}(7)$ be independent random variables. Define $Z = 3W_1 - 2W_2 + 4$.

A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let $\text{P}(X = n)$ be the probability that Kati obtains her third voucher on the $n$th bar opened.

(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)

It is given that $\text{P}(X = n) = \frac{n^2 + an + b}{2000} \times 0.9^{n - 3}$ for $n \geqslant 3, n \in \mathbb{N}$.

Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.

Linda is a farmer who grows and sells zucchinis. Interested in the weights of zucchinis produced, she records the weights of eight zucchinis and obtains the following results in kilograms.

Assume that these weights form a random sample from a $N(\mu, \sigma^2)$ distribution.

Linda claims that the mean zucchini weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis $H_0: \mu = 7.5$.

Consider $n$ independent random variables $X_1, X_2, \ldots, X_n$ with means $\mu_1, \mu_2, \ldots, \mu_n$ and variances $\sigma_1^2, \sigma_2^2, \ldots, \sigma_n^2$. Let $Y = c_1X_1 + c_2X_2 + \ldots + c_nX_n$, where $c_1, c_2, \ldots, c_n$ are constants.