1. PROBABILITY AND STATISTICS WEEK 4 Onur Doğan

2. Random Variable Random Variable. Let S be the sample space for an experiment. A real-valued function that is defined on S is called a random variable. Onur Doğan

3. Random Variable 1.Discrete Random Variable: Has a finite (or countably infinite) range. Tossing a coin: X= 0 for head and X= 1 for tail 2.Continuous Random Variable: Has an interval of real numbers for its infinite range. The life length of a light bulb: X ≥ 0 Onur Doğan

4. Reminder ! 2.s 2 and s are the variance and standard deviation of the sample 3., s 2, and s are called sample statistics 4.  (lowercase Greek letter “mu”) is the mean of the population 5.  2 (“sigma squared”) is the variance of the population 6.  (lowercase Greek letter “sigma”) is the standard deviation of the population 7. ,  2, and  are called population parameters. (A parameter is a constant. ,  2, and  are typically unknown values.) 1. is the mean of the sample

5. Discrete Random Variables Let 4 coins tossed, and let X be the number of heads that are obtained. Let us find the distributions of that experiment. Onur Doğan

6. Probability Distribution Onur Doğan

7. Discrete Random Variables Example Three balls, a, b, c, are randomly distributed in three boxes. Determine the distribution of the random variable X ="the number of non-empty boxes". Onur Doğan

8. Discrete Random Variables Example Consider a group of five potential blood donors; “a, b, c, d, and e” of whom only a and b have type 0+ blood. Five blood samples, one from each individual, will be typed in random order until an 0+ individual is identified. Let the rv Y=“the number of typings necessary to identify an 0+ individual.” Find the pmf. Onur Doğan

9. The Cumulative Distribution Function Onur Doğan

10. Example Onur Doğan

11. The Expected Value of X (Mean of a Discrete Random Variable) Onur Doğan

12.   [()]x.px The Expected Value of X (Mean of a Discrete Random Variable) The mean, , of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together: Notes: The mean is the average value of the random variable, what happens on average The mean is not necessarily a value of the random variable

13. The Variance of X Onur Doğan

14. Example The grades of n = 50 students in a statistics class are summarized as follows: Find the pmf, mean, variance and sd. Onur Doğan Grade (X) 1234 Number of Students 1020155

15. Example Variance and Standard Deviation of a Discrete Distribution. Suppose that a random variable X can take each of the five values −2, 0, 1, 3, and 4 with equal probability. Determine the variance and standard deviation of X. Onur Doğan

16. A Shortcut Formula for V(X) Onur Doğan

17. Example Determine the mean, variance, and standard deviation of casting a single die (X). Onur Doğan

18. Example Onur Doğan

19. Example A shipment of 8 similar microcomputers to contains 3 defective one. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives. Onur Doğan

20. Example Onur Doğan

21. Example Example:The probability distribution for a random variable x is given by the probability function: Find the mean, variance, and standard deviation

22. Discrete Uniform Distribution A discrete uniform random variable X has an equal probability for each value in the range of X= [a, b], a < b. Thus, the probability mass function of X is; P(x)= 1/(b-a+1) where x=a,a+1,…,b Onur Doğan

23. Example Casting a die… Onur Doğan

24. Example Suppose that product codes of 2, 3, or 4 letters are equally likely. Determine the probability mass function of the number of letters (X) in a product code. Calculate the mean and variance of X Onur Doğan