1. Random Variable (RV) A function that assigns a numerical value to each outcome of an experiment. Notation: X, Y, Z, etc Observed values: x, y, z, etc

2. Example 1 Two fair coins tossed. Let X = No of Heads OutcomesProbabilityValue of X HH¼2 HT¼1 TH¼1 TT¼0

3. Random Variable Discrete RV – finite or countable number of values Continuous RV – taking values in an interval

4. Probability Distribution Probability distribution of a discrete RV described by what is known as a Probability Mass Function (PMF). Probability distribution of a continuous RV described by what is known as a Probability Density Function (PDF).

5. Probability Mass Function (PMF) p(x) = Pr (X = x) satisfying p(x) ≥ 0 for all x ∑p(x) = 1

6. Example 1 (Contd) X = No of Heads. The PMF of X is: xPr (X = x) 01/4 12/4=1/2 21/4

7. Example 1 (Contd)

8. Probability Density Function (PDF) f(x) satisfying f(x) ≥ 0 for all x ∫f(x) dx = 1 ∫ a b f(x) dx = Pr (a < X < b) Pr (X = a) = 0

9. Example 2

10. Example 3

11. Expectation E (X) = ∑x p (x) for a Discrete RV E (X) = ∫x f (x) dx for a Continuous RV

12. Expectation E (X 2 )= ∑x 2 p (x) for a Discrete RV E (X 2 ) = ∫x 2 f (x) dx for a Continuous RV

13. Expectation E (g(X)) = ∑g(x) p (x) for a Discrete RV E (g(X)) = ∫g(x) f (x) dx for a Continuous RV

14. Variance Var (X) = E (X 2 ) – (E(X)) 2

15. Standard Deviation SD (X) = √Var (X)

16. Properties of Expectation E (c) = c for a constant c E (c X) = c E (X) for a constant c E (c X + d) = c E (X) + d for constants c & d

17. Properties of Variance Var (c) = 0 for a constant c Var (c X) = c 2 Var (X) for a constant c Var (c X + d) = c 2 Var (X) for constants c & d

18. Example 1 (Contd) X = No of Heads. Find the following: (a)E (X)Ans:1 (b)E (X 2 )Ans:1.5 (c)E ((X+10) 2 )Ans: 121.5 (d)E (2 X )Ans:2.25 (e)Var (X)Ans:0.5 (f)SD (X)Ans:1/√2

19. Example 4 If X is a random variable with the probability density function f (x) = 2 (1 - x) for 0 < x < 1 find the following: (a)E (X)Ans:1/3 (b)E (X 2 )Ans:1/6 (c)E ((X+10) 2 )Ans:106.8333 (d)Var (X)Ans:1/18 (e)SD (X)Ans:1/(3 √ 2)

20. Example 5 An urn contains 4 balls numbered 1, 2, 3 & 4. Let X denote the number that occurs if one ball is drawn at random from the urn. What is the PMF of X?

21. Example 5 (Contd) Two balls are drawn from the urn without replacement. Let X be the sum of the two numbers that occur. Derive the PMF of X.

22. Example 6 The church lottery is going to give away a £3,000 car and 10,000 tickets at £1 a piece. (a)If you buy 1 ticket, what is your expected gain. (Ans: -0.7) (b)What is your expected gain if you buy 100 tickets? (Ans: -70) (c)Compute the variance of your gain in these two instances. (Ans: 899.91 & 89100)

23. Example 7 A box contains 20 items, 4 of them are defective. Two items are chosen without replacement. Let X = No of defective items chosen. Find the PMF of X.

24. Example 8 You throw two fair dice, one green and one red. Find the PMF of X if X is defined as: A) Sum of the two numbers B) Difference of the two numbers C) Minimum of the two numbers D) Maximum of the two numbers

26. Example 9 If X has the PMF p (x) = ¼ for x = 2, 4, 8, 16 compute the following: (a)E (X)Ans:7.5 (b)E (X 2 )Ans:85 (c)E (1/X)Ans: 15/64 (d)E (2 X/2 )Ans:139/2 (e)Var (X)Ans:115/4 (f)SD (X)Ans:√115/2

27. Example 10 If X is a random variable with the probability density function f (x) = 10 exp (-10 x) for x > 0 find the following: (a)E (X)Ans:0.1 (b)E (X 2 )Ans:0.02 (c)E ((X+10) 2 )Ans:102.02 (d)Var (X)Ans:0.01 (e)SD (X)Ans:0.1

28. Example 11 If X is a random variable with the probability density function f (x) = (1/ √ (2  )) exp (-0.5 x 2 ) for -  < x <  find the following: (a)E (X)Ans:0 (b)E (X 2 )Ans:1 (c)E ((X+10) 2 )Ans:101 (d)Var (X)Ans:1 (e)SD (X)Ans:1

29. Example 12 A game is played where a person pays to roll two fair six-sided dice. If exactly one six is shown uppermost, the player wins £5. If exactly 2 sixes are shown uppermost, then the player wins £20. How much should be charged to play this game is the player is to break-even?

30. Example 13 Mr. Smith buys a £4000 insurance policy on his son’s violin. The premium is £50 per year. If the probability that the violin will need to be replaced is 0.8%, what is the insurance company’s gain (if any) on this policy?