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

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

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. 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. 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. 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. 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. 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. ; { x1, x2, …} = all possible X values • pi is the probability X = xi where i = 1, 2, … • The expected value of X is:

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

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. 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. Binomial Distribution • p = 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. 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. Mean & Variance of a Binomial R.V. • Notations as before • Mean is • Variance is

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. 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. Finding Probability Y: a normal r.v. with mean m and standard deviation s • Finding z scores • Shade the required area under the standard normal curve • Use Z-Table (p. 1170) to find the answer

17. Example • Suppose that the final scores of ST6304 students follow a normal distribution with m = 80 and s = 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. 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 m=0.9 and variance s2=1.29.

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

21. Sampling Distribution of Sample Mean

22. Sampling Distribution of Sample Mean

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

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

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