Discussion questions · Section 2 of 3

X and Y are independent continuous normal random variables with means 10,12 and standard deviations 2,3 respectively. Find

(a) $P(Y<10)$,

(b) $P(2X>25)$,

(c) $P(Y<X)$,

(d) $P(4X+5Y>90)$,

(e) x such that $P(X_1+X_2>x)=0.25$, for independent observations $X_1,X_2$ of X.

Type A rod lengths are normally distributed with mean 5 cm and standard deviation 0.5 cm; type B lengths have mean 10 cm and standard deviation 1 cm. Find the probability that

(a) the total length of 3 type A rods and 2 type B rods lies between 33 and 36 cm,

(b) the total length of 6 type A rods exceeds three times the length of one randomly chosen type B rod.

Pineapple masses in a large warehouse are normally distributed with mean 1.52 kg and standard deviation 0.070 kg. Six randomly chosen pineapples are packed in a box of mass 2.15 kg. Find the probability that the filled box has mass below 11.2 kg.

For 3 randomly chosen filled boxes, find the probability that at least 1 has mass below 11.2 kg.

The contents of a packet of cereal A have a normal mass distribution with mean 625 g and standard deviation 15 g. Packaging has an independent normal distribution with mean 25 g and standard deviation 3 g. Give answers to 3 significant figures.

(a) Find the probability that one packet of A has total mass exceeding 630 g.

(b) Find the probability that the contents of 4 random packets of A exceed 2450 g.

Cereal B contents have normal mass with mean 465 g and standard deviation 10 g. Find the probability that the contents of 4 random packets of B outweigh the contents of 3 random packets of A.

Chicken and turkey masses in kg have independent normal distributions.

Bird	Mean mass	Standard deviation
Chicken	2.2	0.5
Turkey	10.5	2.1

Chickens cost 3 dollars per kg and turkeys 5 dollars per kg.

(i) Find the probability that a random chicken costs more than 7 dollars.

(ii) Find the probability that both a random chicken costs more than 7 dollars and a random turkey more than 55 dollars.

(iii) Find the probability that their total price exceeds 62 dollars.

(iv) Explain why(iii) is greater than (ii).

Independent X,Y have distributions $N(2\mu,\sigma^2)$ and $N(3\mu,3\sigma^2)$ respectively, with σ>0. Let $Z\sim N(0,1)$.

(i) Given $P(X<2\mu+2\sigma^2)=k$, show that $P(Z<2\sigma)=k$.

(ii) Write down the mean and variance of Y−X. Hence find $P(|Y-X-\mu|>4\sigma^2)$ in terms of k.