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

2 Random Variables Section 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. ; { 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 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  = 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  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 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 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.

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)

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

Sampling Distribution of Sample Mean 21

Sampling Distribution of Sample Mean 22

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

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

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

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