1. week71 Discrete Random Variables A random variable (r.v.) assigns a numerical value to the outcomes in the sample space of a random phenomenon. A discrete r.v X has a finite number of possible values. The probability distribution of X lists the values x i and their probabilities p i. Every p i is a number between 0 and 1. The sum of the p i ’s must equal 1. Examples 1. Consider the experiment of tossing a coin. Define a random variable as follows: X = 1 if a H comes up = 0 if a T comes up. This is an example of a Bernoulli r.v. The Probability function of X is given in the following table xP(X = x) 1010 p 1- p

2. week72 2.Let X be a r.v counting the number of girls in a family with 3 children. The probability function of X is given in the following table. 3.Toss a coin 4 times. Let X be the number of H’s. Find the probability function of X. Draw a probability histogram. 4.Toss a coin until the 1st H. Let X be the number of T’s before the 1st H. Find the probability function of X. xP(X = x) 01230123 (0.5) 3 = 0.125 3·(0.5) 3 = 0.375 (0.5) 3 = 0.125

3. week73 Continuous random variables A continuous r. v. X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The total area under a density curve is 1. The probability of any event is the area under the density curve and above the value of X that make up the event. Example The density function of a continuous r. v. X is given in the graph below. Find i) P(X < 7) ii) P(6 < X < 8) iii) P(X = 7) iv) P(5.5 < X < 7 or 8 < X < 9)

4. week74 Normal distributions The density curves that are most familiar to us are the normal curves.

5. week75 Mean (expected value) of a discrete r. v. The mean of a r. v. X is denoted by μ x and can be found using the following formula: Examples: 1. The mean of the Bernoulli r.v defined in example 1 above is : μ x = 0·(1-p) + 1·p = p 2. The mean number of girls in a family with 3 children is 1.5. Exercise: Find the mean of X in example 3 above. Exercise: Read Example 4.20 on p291 in IPS.

6. week76 Example – Discrete Uniform r.v Roll a six-sided die. Define a r. v. X to be the number shown on the die. That is, X = 1 if die lands on 1, X = 2 if die lands on 2, etc. The probability distribution of X is given in the table below: The mean of X is μ X = 1·(1/6) + 2·(1/6) + 3·(1/6) + 4·(1/6) +5·(1/6) + 6·(1/6) = 21/6 = 3.5. xP(X = x) 123456123456 1/6

7. week77 Law of large numbers If independent observations are drawn from a population with a finite mean , the population mean  can be estimated with a specified degree of accuracy by the sample mean, using sufficiently large sample.

8. week78 Rules for Mean of r.v For any two r.v’s X and Y and constants a and b, 1. μ x + a = μ x + a. 2. μ b·x = b·μ x. 3. μ b·x + a = b·μ x + a. 4. μ X + Y = μ X + μ Y. Example: The price X of Nike sports shoes is a random variable with mean μ x = 200$. Before the holidays Nike company had a promotion: “ Pay 10$ less for each item and get 20% discount from the original price”. a)What is the mean price during the promotion. b)Suppose in addition that during the promotion the mean price for Nike socks is μ Y = 20$. What is the expected value of your expenses if you are to buy one pair of shoes and one pair of socks ?

9. week79 The variance of a r. v. The variance of a r. v. is an average of the squared deviations from the mean, (X – μ x ) 2. The Variance of a discrete r. v. is The standard deviation σ X of a r. v. is the square root of its variance. Examples: 1.The variance of a Bernoulli r.v is σ 2 x = p – p 2 = p(1- p) 2. The variance of the Uniform example above is σ 2 x = (91/6) – (3.5) 2 = 2.9167

10. week710 Rules for variances If X is a r. v. and a and b are constants, then 1. 2. 3. 4. If X and Y are independent r. v’s then,

11. week711 Two random variables X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the occurrence of any event involving Y alone. Example: Consider again the Nike example above. If the stdev. of X is σ x = 10$ and the stdev. of Y is σ Y = 8$. a)What is the stdev. of the shoes price during the promotion? (8). b)What is the stdev. of your expenses if you were to buy one pair of shoes and one pair of socks ? (12.806). Example 4.34 on page 283 in IPS.