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Problems. Problems 4.17, 4.36, 4.40, (TRY: 4.43). 4. Random Variables. A random variable is a way of recording a quantitative variable of a random experiment. 4. Random Variables. A random variable is a way of recording a quantitative variable of a random experiment.

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problems
Problems

Problems 4.17, 4.36, 4.40, (TRY: 4.43)

4 random variables
4. Random Variables

A random variable is a way of recording a quantitative variable of a random experiment.

4 random variables1
4. Random Variables

A random variable is a way of recording a quantitative variable of a random experiment.

This variable has a distribution, mean and standard deviation, so we can discuss outliers using the same procedures as back in Chapter 2.

4 random variables2
4. Random Variables

A random variable is a way of recording a quantitative variable of a random experiment.

This variable has a distribution, mean (expected value) and standard deviation, so we can discuss outliers using the same procedures as back in Chapter 2.

This includes percentiles, Chebyshev’s Rule and the Empirical Rule!

4 random variables3
4. Random Variables

…outliers using the same procedures as back in Chapter 2.

This includes percentiles, Chebyshev’s Rule and the Empirical Rule!

The difference in this Chapter is we talk about the probabilities of what is to occur and in Chapter 2 we talked about the frequency of what did occur.

4 random variables4
4. Random Variables

The difference in this Chapter is we talk about the probabilities of what is to occur and in Chapter 2 we talked about the frequency of what did occur.

In Chapter 2 we are talking about the sample and in Chapter 4 we are talking about the population.

example
Example

Find the probability distribution obtained by flipping an unbiased coin three times and counting the number of times heads comes up.

binomial experiment
Binomial Experiment

A binomial experiment is one that:

1) Has a fixed number of trials (n)

2) These trials are independent

3) Each trial must have all outcomes classified into two categories (Success or Failure)

4) The probability of success remains constant for all trials.

notation
Notation:
  • S = success and P(S) = p
  • F = Failure and P(F) = q = 1- p
  • n = fixed number of trials
  • x = specific number of successes in n trials
  • P(x) = the probability of getting exactly x successes among n trials
example1
Example

Shaquille Rashaun O'Neal(Shaq) is a basketball player who takes a lot of free throws. The probability of Shaq making a free throw is 0.60 on each throw.

With 3 free throws what is the probability that he makes 2 shots?

notation1
Notation:
  • S = success and P(S) = .6
  • F = Failure and P(F) = .4
  • n = 3
  • x = 2
  • P(2) = the probability of getting exactly 2 successes (successful free throws) among n=3 trials
factorials
Factorials

0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4* 3 * 2 * 1

n! = n*(n-1)!

factorials1
Factorials

0! = 1

1! = 1

2! = 2 * 1=2

3! = 3 * 2 * 1=6

4! = 4* 3 * 2 * 1=24

n! = n*(n-1)!

binomial probability distribution
Binomial Probability Distribution

In a binomial experiment, with constant probability p of success at each trial, the probability of x successes in n trials is given by

example2
Example

Shaq is a basketball player who takes a lot of free throws. The probability of Shaq making a free throw is 0.60 on each throw.

With 3 free throws what is the probability that he makes 2 shots?

example3
Example

Shaq is a basketball player who takes a lot of free throws. The probability of Shaq making a free throw is 0.60 on each throw.

With 3 free throws what is the probability that he makes 2 shots?

example4
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example5
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example6
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example7
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example8
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example9
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

how to use the binomial tables
How to use the Binomial Tables
  • (see page 885)
  • First find the appropriate table for the particular value of n
  • then find the value of p in the top row
  • Find the row corresponding to k and find the intersection with the column corresponding to the value of p
  • The value you obtain is the cumulative probability, that is P(x ≤ k)
  • N=10, p = 0.7: P(x ≤ 4) = 0.047
  • N=10, p = 0.7: P(x = 4) = P(x ≤ 4) - P(x ≤ 3) = 0.047-0.011=0.036
  • N=10, p = 0.7: P(x > 4) = 1- P(x ≤ 4)

= 1 - 0.047 = 0.953

example10
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example11
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

example12
Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

problems1
Problems

Problems 4.52, 4.56, 4.62, 4.64, 4.66, 4.68

keys to success
Keys to success

Learn the binomial table.

Be able to recognize binomial distributions and when you do apply the appropriate formulas and tables.

homework
Homework
  • Review Chapter 4.4
  • Read Chapter 5.1-5.3