Download Lecture 5 - Statistics

Statistics 400 - Lecture 5
 Last class: Finished 4.1-4.4 and started 4.5
 Today: Finish 4.5 and begin discrete random variables (5.1-5.5)
 Next Day: More discrete random variables (5.5-5.7) and begin
continuous R.V.’s (6.1-6.3)
 Assignment #2: 4.14, 4.24, 4.41, 4.61, 4.79, 5.13(a and c), 5.32,
5.68, 5.80
 Due in class Tuesday, October 2
Independent Events
 Two events are independent if:
P( A | B)  P( A)
 The intuitive meaning is that the outcome of event B does not
impact the probability of any outcome of event A
 Alternate form:
P( A and B)  P( A) P( B)
Example
 Flip a coin two times
 S=
 A={head observed on first toss}
 B={head observed on second toss}
 Are A and B independent?
Example
 Mendel used garden peas in experiments that showed inheritance
occurs randomly
 Seed color can be green or yellow
 {G,G}=Green otherwise pea is yellow
 Suppose each parent carries both the G and Y genes
 M ={Male contributes G}; F ={Female contributes G}
 Are M and F independent?
Example (Randomized Response Model)
 Can design survey using conditional probability to help get honest
answer for sensitive questions
 Want to estimate the probability someone cheats on taxes
 Questionnaire:
 1. Do you cheat on your taxes?
 2. Is the second hand on the clock between 12 and 3?

YES
NO
 Methodology: Sit alone, flip a coin and if the outcome is heads
answer question 1 otherwise answer question 2
More on Probability
 Will take a more formal look at describing random phenomenon
 A random variable, X, associates a numerical value to each
outcome of an experiment
 Will consider two types:
 Discrete random variables
 Continuous random variables
Discrete versus Continuous
 Discrete random variables have either a finite number of values or
infinitely many values that can be ordered in a sequence
 Continuous random variables take on all values in some interval(s)
Examples
 Discrete or continuous
 Number of people arriving in a supermarket
 Hair color of randomly selected people
 Weight lost from a diet program
 Random number between 0 and 4
Discrete Random Variables
 Describe chances of observing values for a discrete random variable
by probability distribution
 Probability distribution of a discrete random variable, X, is the list of
distinct numerical outcomes and associated probabilities
Value of X
Probability f(xi)
x1
f(x1)
x2
…
f(x2) …
xk
f(xk)
 If distribution is estimated from data, it is called the empirical
distribution
Properties
 f (x ) 
i
k
  f (x ) 
i 1
i
for each value xi of X
 Can display distribution using a probability histogram
 X-axis represents outcomes
 Y-axis is the probability of each outcome
 Use rectangles, centered at each value of X, to display probabilities
Example
 Probability distribution for number people in a randomly selected
household
X=# people
f(xi)
1
2
3
4
5
6
7
0.25 0.32 0.17 0.15 0.07 0.03
 Draw the probability histogram
Mean and Variance for Discrete
Random Variables
 Suppose have 1000 people in a population (500 male and 500
female) and average age of the males is 26 and average age of
females is 24
 What is the mean age in the population?
 Suppose have 1000 people in a population (900 male and 100
female) and average age of males is 26 and average age of females
is 24
 What is the mean age in the population?
 Mean must consider chance of each outcome
 Mean is not necessarily one of the possible outcomes
 Is a weighted average of the outcomes
 The mean (or expected value) of a discrete R.V., X, is denoted E(X)
k
E( X )   f ( x ) x
i 1
 Is also denoted as 
   E (X )
i
i
 Variance of a discrete R.V. weights the squared deviations from the
mean by the probabilities
  Var ( X )   ( x   ) f ( x )
k
2
i 1
2
i
i
 The standard deviation is
   (x  ) f (x )
k
i 1
2
i
i
Example
 Compute mean and variance of number of people in a household
Example (true story)
 People use expectation in real life
 Parking at Simon Fraser University (B.C., Canada) is $9.00 per day
 Fine for parking illegally is $10.00
 When parking illegally, get caught roughly half the time
 Should you pay the $9.00 or risk getting caught?
 Probability Model - is an assumed form of a distribution of a
random variable
Bernoulli Distribution
 Bernoulli distribution:
 Each trial has 2 outcomes (success or failure)
 Prob. of a success is same for each trial
 Prob. of a success is denoted as p
 Prob. of a failure, q, is
 Trials are independent
 If X is a Bernoulli random variable, its distribution is described by
f ( x)  P( X  x)  p x q1 x
 where X=0 (failure) or X=1 (success)
Example
 A backpacker has 3 emergency flares, each which light with
probability of 0.98.
 Find probability the first flare used will light
 Find probability that first 2 flares used both light
 Find probability that exactly 2 flares light
Mean and Standard Deviation
 Mean:
 Standard Deviation: