1. 1 Since everything is a reflection of our minds, everything can be changed by our minds.

2. 2 Random Variables Section 4.6-4.14 Types of random variables Binomial and Normal distributions Sampling distributions and Central limit theorem Random sampling Normal probability plot

3. 3 What Is a Random Variable? A random variable (r.v.) assigns a number to each outcome of a random circumstance. Eg. Flip two coins: the # of heads When an individual is randomly selected and observed from a population, the observed value (of a variable) is a random variable.

4. 4 Types of Random Variables A continuous random variable can take any value in one or more intervals. We cannot list down (so uncountable) all possible values of a continuous random variable. All possible values of a discrete random variable can be listed down (so countable).

5. 5 Distribution of a Discrete R.V. X = a discrete r.v. x = a number X can take The probability distribution of X is: P(x) = P(Y=x)

6. 6 How to Find P(x) P(x) = P(X=x) = the sum of the probabilities for all outcomes for which X=x Example: toss a coin 3 times and x= # of heads

7. 7 Expected Value (Mean) The expected value of X is the mean (average) value from an infinite # of observations of X. X = a discrete r.v. ; { x 1, x 2, …} = all possible X values p i is the probability X = x i where i = 1, 2, … The expected value of X is:

8. 8 Variance & Standard Deviation Variance of X: Standard deviation (sd) of X:

9. 9 Binomial Random Variables Binomial experiments (analog: flip a coin n times): Repeat the identical trial of two possible outcomes (success or failure) n times independently The # of successes out of the n trials (analog: # of heads) is called a binomial random variable

10. 10 Example Is it a binomial experiment? Flip a coin 2 times The # of defective memory chips of 50 chips The # of children with colds in a family of 3 children

11. 11 Binomial Distribution  = the probability of success in a trial n = the # of trials repeated independently Y = the # of successes in the n trials For y = 0, 1, 2, …,n, P(y) = P(Y=y)= Where

12. 12 Example: Pass or Fail Suppose that for some reason, you are not prepared at all for the today’s quiz. (The quiz is made of 5 multiple-choice questions; each has 4 choices and counts 20 points.) You are therefore forced to answer these questions by guessing. What is the probability that you will pass the quiz (at least 60)?

13. 13 Mean & Variance of a Binomial R.V. Notations as before Mean is Variance is

14. 14 Distribution of a Continuous R.V. The probability distribution for a continuous r.v. Y is a curve such that P(a < Y <b) = the area under the curve over the interval (a,b).

15. 15 Normal Distribution The most common distribution of a continuous r.v.. The normal curve is like: The r.v. following a normal distribution is called a normal r.v.

16. 16 Finding Probability Y: a normal r.v. with mean  and standard deviation  1. Finding z scores 2. Shade the required area under the standard normal curve 3. Use Z-Table (p. 1170) to find the answer

17. 17 Example Suppose that the final scores of ST6304 students follow a normal distribution with  = 80 and  = 5. What is the probability that a ST6304 student has final score 90 or above (grade A)? Between 75 and 90 (grade B)? Below 75 (Fail)?

18. 18 Sampling Distribution A parameter is a numerical summary of a population, which is a constant. A statistic is a numerical summary of a sample. Its value may differ for different samples. The sampling distribution of a statistic is the distribution of possible values of the statistic for repeated random samples of the same size taken from a population.

19. Sampling Distribution of Sample Mean Example: suppose the pdf of a r.v. X is as follows: Its mean  and variance    19 x013 f(x)0.50.30.2

20. Sampling Distribution of Sample Mean 20 All possible samples of n=2:

21. Sampling Distribution of Sample Mean 21

22. Sampling Distribution of Sample Mean 22

23. 23 Central Limit Theorem When n is large, the distribution of y is approximately normal.

24. 24 Central Limit Theorem (uniform[0,1])

25. Normal Approximation to Binomial Distribution The binomial distribution is approximately normal when the sample size is large enough: Continuity correction 25

26. Others Random sampling and Normality checking are in Lab 2 Poisson Distribtion 26