1. Discrete Random Variables To understand what we mean by a discrete random variable To understand that the total sample space adds up to 1 To understand the P(X=x) notation To use the P(X=x) notation to solve problems

2. Discrete Random Variables Value is from an experiment in the real world The value is numerical X is a random variable (e.g. X = Heads on a coin) x is a particular variable (e.g. 1 head in 2 throws) Possible outcomes can be shown in a sample space P(X=x) would be the probability of throwing 1 head in 2 throws of a coin

3. Are these Discrete Random Variables? The average lifetime of a light bulb Not discrete, as time is continuous The number of days in January No, not variable as there are always 31 The number of moves it takes to win a game of draughts Yes, as number of moves are whole numbers and it varies game by game

4. Sample Space 3 coins are tossed and the number of heads, X, are recorded a)Show the sample space b)Write down the probability distribution c)Write down the probability function Sample space HHH, THH, HTH, HHT, TTH, THT, HTT, TTT

5. Probability distribution 3 coins are tossed and the number of heads, X, are recorded a)Show the sample space b)Write down the probability distribution c)Write down the probability function Sample space HHH, THH, HTH, HHT, TTH, THT, HTT, TTT x0123 P(X=x) Note that the probabilities add up to 1

6. Probability distribution 3 coins are tossed and the number of heads, X, are recorded a)Show the sample space b)Write down the probability distribution c)Write down the probability function Sample space HHH, THH, HTH, HHT, TTH, THT, HTT, TTT x0123 P(X=x) P(X=x) =, for x = 0,3, x = 1,2 0, otherwise

7. Example A tetrahedral die is numbered 1,2,3,4. The die is biased. P(die landing on any number = k / x where k is a constant. a)Find the value of k b)Write down the probability distribution for X, the number the die lands on after a single roll x1234 P(X=x) K/1K/1 K/2K/2 K/3K/3 K/4K/4 K / 1 + K / 2 + K / 3 + K / 4 = 1 12k + 6k + 4k + 3k = 1 12 25k = 1 12 25k = 12 k = 12 / 25

8. Example A tetrahedral die is numbered 1,2,3,4. The die is biased. P(die landing on any number = k / x where k is a constant. a)Find the value of k b)Write down the probability distribution for X, the number the die lands on after a single roll x1234 P(X=x) 12 / 25 12 / 50 12 / 75 12 / 100