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

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Random Variables

Section 4.6-4.14Types of random variablesBinomial and Normal distributionsSampling distributions and Central limit theoremRandom samplingNormal probability plot

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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.

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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).

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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)

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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

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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 valuespi is the probability X = xi where i = 1, 2, …The expected value of X is:

i

ii pxXE )(

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Variance & Standard Deviation

Variance of X:

Standard deviation (sd) of X:

i

ii pxXV 22 )()(

i

ii px 2)(

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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

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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

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Binomial Distribution

= the probability of success in a trial n = the # of trials repeated independently Y = the # of successes in the n trialsFor y = 0, 1, 2, …,n, P(y) = P(Y=y)=

Where

yny

yny

n

)1()!(!

!

1)...2)(1(! nnnn

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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)?

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Mean & Variance of a Binomial R.V.

Notations as before

Mean is

Variance is

n

)1(2 n

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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).

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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.

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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

y

z

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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)?

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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

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x 0 1 3

f(x) 0.5 0.3 0.2

Sampling Distribution of Sample Mean

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All possible samples of n=2:

Sampling Distribution of Sample Mean

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Sampling Distribution of Sample Mean

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Central Limit Theorem

nyy / and

When n is large, the distribution of y is approximately normal.

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

Normal Approximation to Binomial Distribution

The binomial distribution is approximately normal when the sample size is large enough:

Continuity correction

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5;5 nn

Others

Random sampling and Normality checking are in Lab 2

Poisson Distribtion

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