Download STAB22 Statistics I Lecture 17 1

STAB22 Statistics I
Lecture 17
1
Random Variables

Random Variable (RV) takes single value for
every outcome of an experiment X H ,H   2 X H ,T   1


Eg. X = # of heads in 2 fair coin flips XT ,H   1 XT ,T   0
Probability Model describes all possible
values & related probabilities of RV

Centre of RV (Expected Value or Mean)
  E  X    xP  x 
x
P(x)
0
¼
1
½
2
¼
Sum
1
all x

Spread of RV (Variance & Std. Deviation)
2
2
  Var ( X )    x    P  x  &   SD (  )  Var ( X )
all x
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Example


In roulette, $1 bet on “even” wins $1 if ball
falls on even number; odds of winning are
47.37%
Random Variable X = win/loss of $1 bet

What are the possible values of X?

What is the distribution of X?

What is the expected value of X?
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Linear Transformations


Often need to change values of RV
E.g. Let X = # of Heads in X   2 X   1
X    1 X   0
2 fair coin flips
Gamble offers $4 for every Head, but have to
pay $5 to play. Let Y = # net gamble gain



If X=0, Y=−5+4∙0=−5
If X=1, Y=−5+4∙1=−1
If X=2, Y=−5+4∙2=+3
→ Generally: Y = −5 + 4∙X
H ,H
H ,T
T ,H
T ,T
X value
Y value
Prob
0
−5
¼
1
−1
½
2
+3
¼
Sum
(change only RV values, not probs)
1
4
Centre & Spread of Linear
Transformations

Don’t need to calculate centre & spread of
linear transformation RV from scratch
If Y = a∙X + b (for any a, b), then
E Y   E  a  X  b   a  E  X   b
Var Y   Var  a  X  b   a 2 Var  X 
 SD Y  | a | SD  X 

E.g. For previous gamble E(X)=1 →
→ E(Y) = E(4∙X − 5) = 4∙E(X) − 5 = 4 − 5 = −1
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Example

Tomorrow’s temperature in °C is RV X with
E(X) = 5, SD(X) = 2. Find mean & st.dev. of
tomorrow’s temperature in °F

Note: [°F] = [°C] · 9/5 + 32
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Combining Random Variables


Often, we need to combine RV’s
E.g. Let X = # Heads in two coin flips,
and Y = # Heads in another coin flip
Define Z = X + Y (# Heads in all 3 flips)

Find probability model of Z by considering all
possible combinations of X + Y
x
P(x)
0
¼
1
½
2
¼
z
&
y
P(y)
0
½
1
½
P(z)
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Centre & Spread of
Combination of RV’s

Again, don’t need to calculate centre &
spread of combination of RV’s from scratch
For any two RV’s X, Y
E  X  Y   E  X   E Y 

For two independent RV’s X, Y
Var  X  Y   Var  X   Var Y 
SD  X  Y   Var  X   Var Y 

Note: RV’s are independent if value of one does
not affect probabilities of the other
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Combinations of Normal RV’s

For Normal & independent RV’s in particular,
means & st. dev.’s are all we need to know to
calculate probabilities of combinations
If X1 is Normal(µ1,σ1), X2 is Normal(µ2,σ2),
and they are independent, then

 X 1  X 2  is Normal 1  2 ,

 12   22

Use this result to calculate probabilities of
combination using Normal distribution
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Example

John & Mary will run a 5K race; their times
are indep. Normal w/ means of 30 & 35min,
and st. dev.’s of 3 & 4min, respectively

Find prob. that their total time is <60min

Find prob. that Mary is faster
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Bernoulli Trial

Bernoulli Trial: trial with only 2 outcomes

E.g. True/False, Yes/No, Heads/Tails



P(Success) = p, P(failure) = 1−p
Value
1
0
Prob
p
1‒p
E.g. Fair coin flip, let Success = Heads


Usually labeled Success (1) and Failure (0)
P(1) = ½, & P(0) = 1− ½ = ½
Bernoulli trials form basis of many common
probability models
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Binomial Model

Several Bernoulli trials, but only interested in
total number of successes


Example: # students who vote Yes for a proposal
Binomial Setting:
1. Fixed number (n ) of Bernoulli trials
2. Same probability of success (p ) for each trial
3. Bernoulli trials are independent

Binomial Random Variable:

X = # of successes in a Binomial setting
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Binomial Distribution

If X follows Binomial distribution with n trials
and probability of success p


X takes values from 0 to n (i.e. 0,1,…,n)
Probabilities given by formula
P ( X  x)  n Cx p x (1  p ) n  x , for x  0,1,..., n
n!
where: n Cx 
, n !  n  (n  1)   2 1
x ! n  x  !

Or, simply use software
(StatCrunch: Stat > Calculators > Binomial)
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Example

Multiple choice test has 10 questions, each with 4
choices: A,B,C or D.
Student has not studied at all, but thinks he will
give it a shot (i.e. answer at random).
What is the probability model of his score?

Does it fit the Binomial?




Number of trials?
Are they independent?
Probability of success?
Is the score a binomial RV?
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Binomial Example

Find probability student’s test score is 5/10

Find probability student passes (score ≥5/10)
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