1. PROBABILITY DISTRIBUTIONS Examples: Sometimes, quantitative variables have values which are based on chance or random outcomes. In this case, they are called RANDOM VARIABLES. W = Time it takes for a randomly selected student to answer a given Math 2N test X = Number of bad checks drawn on Metrobank on a randomly selected day Y = Amount of gasoline needed to drive certain car for 50 miles on a randomly selected route Z = Number of registered voters in a randomly selected sample of 50 adults. A random variable is usually denoted as X.

2. Recall: A set of data values (of a given variable) can be arranged into a frequency distribution or a relative freq. distribution. The relative frequency distribution of a random variable is also called its PROBABILITY DISTRIBUTION. Example: Dr. Reyes administered a boredom tolerance test to a group of 20000 secondary students. The scores were 0, 1, 2, 3, 4, 5, 6 – 6 being the highest tolerance for boredom. ScoreNo. of students 01400 12600 23600 36000 44400 51600 6400

3. Let X = Score of a randomly selected student in the sample of 20,000 students given the boredom tolerance test Score, XP(X=?) 0 1 2 3 4 5 6 Then the probability distribution of the random variable X is: 1400/20000 = 0.07 2600/20000 = 0.13 3600/20000 = 0.18 6000/20000 = 0.3 4400/20000 = 0.22 1600/20000 = 0.08 400/20000 = 0.02 What is P(X=2)? What is P(X=5 or X=6)? What is P(X=3 or X=4)?

4. Frequency Distribution and Graph (Histogram) ScoreNo. of students 01400 12600 23600 36000 44400 51600 6400

5. Probability Distribution and Graph (Histogram) Score, XP(X=?) 00.07 10.13 20.18 30.3 40.22 50.08 60.02 For any given value of the random variable (ex: X=4) the proba- bility for that value ( P(X=4)=0.22 ) is the area of the rectangle at that value. (Assuming that the width of the rect. is 1 unit) The total area under a probability distribution is 1

6. Mean and Standard Deviation of a Probability Distribution Given a random variable X. The MEAN of the prob. dist. of X is: (also called the EXPECTED VALUE) And the standard deviation is: Given a random variable X. The MEAN of the prob. dist. of X is: (also called the EXPECTED VALUE) And the standard deviation is:

7. Score, XP(X=?) 00.07 10.13 20.18 30.3 40.22 50.08 60.02 Given the probability distribution of the random variable X is: Example: Find the mean of the probability distribution. Find the stand. Deviation of the probability distribution.

8. Example: The number of suits sold per day at a retail store is about 19 to 23. The store manager summarized the sales records over 150 days in the following table: No. of suits soldNo. of days 1920 30 2140 2235 2325 Let X = No. of suits sold in a randomly selected day of 150 days of observation A. Construct a probability distribution table for X. B. Draw the graph (histogram) of the prob. dist. for X. C. What is P(X=22 or X=23)? D. Compute the mean and standard dev.