1. BUS304 – Probability Theory1 Probability Distribution  Random Variable:  A variable with random (unknown) value. Examples 1. Roll a die twice: Let x be the number of times 4 comes up. x = 0, 1, or 2 2. Toss a coin 5 times: Let x be the number of heads x = 0, 1, 2, 3, 4, or 5 3. Same as experiment 2: Let’s say you pay your friend $1 every time head shows up, and he pays you $1 otherwise. Let x be amount of money you gain from the game. What are the possible values of x?

2. BUS304 – Probability Theory2 Discrete vs. Continuous Random variables Random Variables ContinuousDiscrete Examples: Number of students showed up next time Number of late apt. rental payments in Oct. Your score in this coming mid-term exam Examples: The temperature tomorrow The total rental payment collected by Sep 30 The expected lifetime of a new light bulb

3. BUS304 – Probability Theory3 Discrete Probability Distribution Table Graph XP(X) 00.25 10.5 20.25 0 1 2 x.50.25 Probability All the possible values of x

4. BUS304 – Probability Theory4 Exercise  Describe the probability distribution of the following experiments:  Draw a pair of dice, x is the random variable representing the sum of the total points.  In a community with 100 households, 10 do not have kid, 40 have just one kid, 30 have 2 kids, and 20 have 3 kids. Randomly select one household. Let X be the number of kids in the household.

5. BUS304 – Probability Theory5 Measures of Discrete Random Variables  Expected value of a discrete distribution  An weighted average, taking into account the probability  The expected value of random variable x is denoted as E(x) E(x)=  xi P(xi) E(x)= x 1 P(x 1 ) +x 2 P(x 2 ) + … + x n P(x n ) Example:What is your expected gain when you play the flip-coin game twice? x P(x) -2.25 0.50 2.25 E(x) = (-2) * 0.25 + 0 * 0.5 + 2 * 0.25 = 0 Your expected gain is 0! – a fair game.

6. BUS304 – Probability Theory6 Worksheet to compute the expected value  Step1: develop the distribution table according to the description of the problem.  Step2: add one (3rd) column to compute the product of the value and the probability  Step3: compute the sum of the 3rd column  The Expected Value xP(x)x*P(x) -20.25-2*.25=-0.5 00.50*0.5=0 20.252*0.25=0.5 E(x) =-0.5+0+0.5=0

7. BUS304 – Probability Theory7 Exercise  You are working part time in a restaurant. The amount of tip you get each time varies. Your estimation of the probability is as follows:  You bargain with the boss saying you want a more fixed income. He said he can give you $62 per night, instead of letting you keep the tips. Would you want to accept this offer? $ per nightProbability 500.2 600.3 700.4 800.1

8. BUS304 – Probability Theory8 More exercise  What is the expected gain if you plan the flip coin game just once?  Three times?  Four Times?  What is the expected number of kid in that community (see the example on page 4)?

9. BUS304 – Probability Theory9 Rule for expected value  If there are two random variables, x and y. Then E(x+y) = E(x) + E(y)  Example: x is your gain from the flip-coin game the first time y is your gain from the flip-coin game the second time x+y is your total gain from playing the game twice. xP(x) 0.5 1 yP(y) 0.5 1 E(x)=0 E(y)=0 x+yP(x+y) -20.25 00.5 20.25 E(x+y)=0

10. BUS304 – Probability Theory10 Measures – variance  Variance: a weighted average of the squared deviation from the expected value. xP(x)x – E(x)(x-E(x)) 2 (x-E(x)) 2 P(x) 500.250-64=-$14(-14) 2 =196196*0.2=39.6 600.3-$4164.8 700.4$63614.4 800.1$1625625.6 Step 1: develop the probability distribution table.Step 2: compute the mean E(x): 50x0.2+60x0.3+70x0.4+80x0.1=64 Step 3: compute the distance from the mean for each value (x-E(x))Step 4: square each distance (x – E(x)) 2 Step 5: weight the squared distance: (x-E(x)) 2 P(x) Step 6: sum up all the weighted square distance. =39.6+4.8+14.4+25.6=84.4

11. BUS304 – Probability Theory11 Variance and Standard deviation  variance  The variance of a random variable has the same meaning as the variance of population  Calculation is the same as calculating population variance using a relative frequency table.  Written as var(x) or  Standard deviation of a random variable:  Same of the population standard deviation  Calculate the variance  Then take the square root of the variance.  Written as SD(x) or e.g. for the example on page 10

12. BUS304 – Probability Theory12 Homework  Problem 4.40  Problem 4.50