Linear combinations of random variables

A company packs chickpeas into small bags and large bags.

The weight of a small bag of chickpeas is normally distributed with mean 500 g and standard deviation 5 g

A random sample of 3 small bags of chickpeas is taken.

Find the probability that the total weight of these 3 bags of chickpeas is between 1490 g and 1530 g

The weight of a large bag of chickpeas is normally distributed with mean 1020 g and standard deviation 20 g

One large bag and one small bag of chickpeas are chosen at random.

Calculate the probability that the weight of the large bag of chickpeas is at least 30 g more than twice the weight of the small bag of chickpeas. Show your working clearly.

The weights of eggs, E grams, follow a normal distribution, N(60, 3²)

The weights of empty small boxes, S grams, follow a normal distribution, N(24, 1.8²)

The weights of empty large boxes, L grams, follow a normal distribution, N(40, 2.1²)

Small boxes of eggs contain 6 randomly selected eggs.

Large boxes of eggs contain 12 randomly selected eggs.

Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.

Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.

The discrete random variable V has probability distribution

Show that the probability generating function of V is

The discrete random variable W has probability generating function

Use calculus to find

E(W)

Var(W)

Given that V and W are independent,

find the probability generating function of X = V + W in its simplest form.

The discrete random variable Y = 2X + 3

Find the probability generating function of Y

Find P(Y = 15)

The random variable X ~ N(5, 0.4²) and the random variable Y ~ N(8, 0.1²)

X and Y are independent random variables.

A random sample of a independent observations is taken from the distribution of X and one observation is taken from the distribution of Y

The random variable W = X₁ + X₂ + X₃ +…+ X_a + bY and has the distribution N(169, 2²) Find the value of a and the value of b

The probability generating function of the random variable X is

where k is a constant.

Find P(X = 2)

Find the probability generating function of W = 2X + 3

The probability generating function of the random variable Y is

Given that X and Y are independent,

find the probability generating function of U = X + Y in its simplest form.

Use calculus to find the value of Var(U)

The weights of a particular type of apple, A grams, and a particular type of orange, R grams, each follow independent normal distributions.

Find the distribution of

A + R

the total weight of 2 randomly selected apples.

A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.

Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams.

From a large number of apples and oranges, Celeste selects m apples and 1 orange at random. The random variable W is given by

where n is a positive integer.

Given that the middle 95% of the distribution of W lies between 1100.08 and 1499.92 grams,

find the value of m and the value of n

Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm. The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm.

Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels.

One large panel and one small panel are selected at random.

Find the probability that the length of the large panel is more than times the length of the small panel.

Rosa needs 1000 cm of fencing. The large panels cost £80 each and the small panels cost £30 each. Rosa’s plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.

Calculate whether or not the expected cost of Rosa’s plan is cheaper than simply buying 14 small panels.

A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic.

The weights, W kg, of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg. The weights, kg, of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg. The random variables W and X are independent.

Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy.

The manufacturer packs n of these wooden toys and 2 n of these plastic toys into the same container. The maximum weight the container can hold is 252 kg.

The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.

Calculate the value of n.