11.03 Variance and standard deviation

At a local fair, in a game that involves rolling a standard die, players can win a prize depending on what they roll. Each player must pay \$3 to play. The prizes are awarded as follows:

• The player wins \$3 if a 1, 3 or 5 is rolled.

• The player wins \$6 if a 4 or 6 is rolled.

• The player wins \$9 if a 2 is rolled.

The table below represents a discrete probability distribution:

In Brad’s toy box, there are 3 toy cars and 4 toy dinosaurs. Each day, for three days, he takes a toy at random and plays with it, and then puts it back.

The notation for standard deviations, \sigma and s, are used to uniquely separate and identify whether the standard deviation reported is a population standard deviation or a sample standard deviation. Which one is which?

A school held a fundraiser to raise money for their annual ski trip, and on average each student in the school raised \$228.

If one class in the school raised an average of \$189 per student, which of the following is possible?

Consider the table:

For each of the following probability distribution tables:

A fair die is rolled. If it lands on an odd number, then the score is the number. But if it lands on an even number, then the score is 0. Let X be the score when the die is rolled once.

Let X be the sum of the numbers that a die lands on when it is rolled twice.

A game is played in which a standard six-sided die is rolled. If it lands on a number other than 1, then the score is that number. If it lands on 1, then a second four-sided dice with numbers 3 to 6 is rolled and the number that die lands on is the score. Let X be the score of a player in this game.

Two spinners numbered from 0 to 4 are spun. Let X be the product of the two numbers that come up.

To open a tutoring business, Quentin does some market research into the profit or loss of similar businesses in his area in the first year. His findings are summarised in the table shown:

Let the random variable X represent the number of students away sick from Year 9 classroom. Similarly, let Y be the random variable for the Year 11 classroom. The probability distributions are shown in the tables below:

Consider the distributions for the random variables W, X, Y and Z.

List the random variables in order from largest variance to smallest variance.

The Sieve of Eratosthenes is a simple algorithm to find all the prime numbers. The prime numbers less than 100 are shown in the diagram below:

We can define a a random variable X on the sample space n = \left\{2, 3, 4, \cdots, 99, 100\right\} modelled by the rule:X(n) = \begin{cases} 0, \text{ if } n \text{ is a composite number }\\\\ 1, \text{ if } n \text{ is a prime number } \end{cases}

X is a random variable for the ages of children in day-care centers across the country and has the following probability distribution:

A die was manufactured such that an odd number is twice as likely to be rolled as an even number. Let X be the number the die lands on.

X is a random variable with the following probability distribution table:

Two soccer players, Sam and Mary, have estimated their probability of scoring a certain number of goals in a game, from their last season's data. Let the random variables S and M be the number of goals scored in a game by Sam and Mary respectively. Their probability distributions are shown in the table below: