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MM Applied Cheat Sheet – AS Stats
Notations, Modelling, Rules and Reminders
Probability ℙ(𝑋); Expectation 𝔼(𝑋); Variance 𝑉𝑎𝑟(𝑋)
𝑋; = ⋯: definition not property
Σ: summation operator
Π: product operator
𝑋 ⊔ 𝑌: X and Y independent
𝑋 ∼ ⋯: X follows …
𝑋: mean; 𝑋̂: estimator (unknown)
𝑛

∑𝑥 =
𝑥=1

𝑥(𝑥 + 1)
2

𝑛

∑ 𝑥2 =
𝑥=1

-

𝑥(𝑥 + 1)(2𝑥 + 1)
6

Draw Venn diagram for set theory
...

𝑥 ∈ [𝑎, 𝑏) means include a exclude b in range
1
√2𝜋

𝑒

−

𝑥2
2

can be written as

1
√2𝜋

𝑥2

exp⁡{− }
2

MM Applied Cheat Sheet – AS Stats
1
...
ℙ(𝐴𝑐 ) = 1 − ℙ(𝐴)
2
...
ℙ(𝐴 ∪ 𝐵) = ℙ(𝐴) + ℙ(𝐵)
4
...
2 Independence
ℙ(𝐴 ∩ 𝐵) = ℙ(𝐴) × ℙ(𝐵)
A family of events 𝒜 = {𝐴𝐼 , 𝑖 ∈ 𝐼} independent: ℙ(𝑛𝐴𝑖 ) = ∏ ℙ (𝐴𝑖 )
A family of events pairwise independent (all combos independent): ℙ(𝐴𝑖 ∩ 𝐴𝑗 ) = ⁡ℙ(𝐴𝑖 )⁡ℙ(𝐴𝑗 )

1
...
4 Partition Theorem
{𝐸𝑖 }𝑖≥1 is a partition of Ω s
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any event cannot happen
together at the same time
...
𝐸𝑖 ∩ 𝐸𝑗 = ∅
2
...
ℙ(𝐴 = 1|𝐵 = 1) =

ℙ(𝐴

= 1|𝐵 = 1)
ℙ(𝐵=1)

ℙ(𝐵 = 1|𝐴 = 1)⁡ℙ(𝐴 = 1)
=
ℙ(𝐵 = 1|𝐴 = 1)⁡ℙ(𝐴 = 1)
+ℙ(𝐵 = 1|𝐶 = 1)⁡ℙ(𝐶 = 1)
…

MM Applied Cheat Sheet – AS Stats
2
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Pre-image of an event, ie
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If domain of X finite/countable subset of ℝ, then RV is discrete
3
...
ie
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Probability mass function of 𝑋 pmf is a function to show the true probability of a random
variable, ie
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2 Expectations
(Or the expected value 𝜇) is actually just the average
...

1
...

3
...


𝔼(ℎ(𝑥)) = ∑𝑛𝑥 ℎ(𝑥)⁡ℙ(𝑋 = 𝑥)
𝔼(𝑎𝑋 + 𝑏) = 𝑎⁡𝔼(𝑥) + 𝑏
𝔼(𝑔(𝑋)) + 𝔼(𝑓(𝑋)) = 𝔼(𝑔(𝑥)) + 𝔼(𝑓(𝑥))
𝔼(𝑔(𝑋)) ≠ 𝑔⁡𝔼(𝑋)

2
...
4 Expectation and Variance of 2 RVs
1
...
𝔼(𝑎𝑋 + 𝑏𝑌) = 𝑎⁡𝔼(𝑋) + 𝑏⁡𝔼(𝑌)
3
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𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌)

MM Applied Cheat Sheet – AS Stats
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2 outcomes
2
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2 Binomial
𝑋~𝐵𝑖𝑛(𝑛, 𝑃):
1
...
of exp
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Each exp
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Each exp
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Independent

Just the Tip™: On calculator,
use Binomial PD for ℙ(𝑋 =
𝑥) and Binomial CD for
ℙ(𝑋 ≤ 𝑥)

ℙ(𝑋 = 𝑥) = 𝑛𝑟𝐶 ⁡𝑃 𝑥 ⁡(1 − 𝑃)1−𝑥
𝔼(𝑋) = 𝑛𝑃
𝑉𝑎𝑟(𝑋) = 𝑛𝑃(1 − 𝑃)

3
...
X = freq
...
Random and independent (events cannot happen at same time)
3
...
4 Uniform
1

ℙ(𝑋 = 𝑥) { 𝑛 ⁡𝑥 = 1⁡𝑡𝑜⁡𝑛
0
1+𝑛
𝔼(𝑋) =
2
𝑛2 − 1
𝑉𝑎𝑟(𝑋) =
12

3
...
𝑋~𝐵𝑒𝑟(𝑃)
2
...
of trials until first success

ℙ(𝑋 = 𝑥) = (1 − 𝑃)𝑥−1 𝑃
1
𝔼(𝑋) =
𝑃
1−𝑃
𝑉𝑎𝑟(𝑋) =
𝑃2

MM Applied Cheat Sheet – AS Stats
4
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State null and hypothesis testing (these are in probability)
2
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Let 𝑋 be …”
3
...
State significance level 𝛼
5
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Calculate the p-value (probability under H0 that test statistic is at least as extreme as observed
value) by subbing in the range of value (inclusive) of 𝑥 that falls under the probability (eg
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Conclusion
a
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If p-value > 𝛼: “insufficient evidence to reject H0”

4
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Consider the probability under H0 and what H1 is (larger/smaller/not H0)
𝛼
6
...
List range of values from 0 to 𝑛
8
...
For two-tailed: each value, find ℙ(𝑝 ≠ 𝑃)using calculator where PD
has X for the value currently testing
9
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1 Introduction to Continuous Random Variables
To find probability of RV which can take uncountably infinite possible values, area can be found by
definition each value not as a line on the graph (where ∫ 𝑓(𝑥) = 1 as total prob ability has to =1)
but as a strip of width 𝛿𝑥
...

c
...
f
...
𝑓𝑋 (𝑥) ≥ 0
∀𝑥 ∈ ℝ
∞
2
...
t
...
d
...
d
...
(probability density function) [equation of curve]
- Not a probability
- Is equation of curve
- Find c
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f
...
d
...
at range b to a where 𝑥 ∈ [𝑎, 𝑏)
𝑑
𝑓𝑋 (𝑥) =
𝐹 (𝑥)
𝑑𝑥 𝑋
At any point for which 𝑓𝑋 (𝑥) is continuous
N
...
: for a point, p
...
f
...
equation of curve is always true for any point), but the point
does not contribute to c
...
f
...
d
...
is easier to find as it is the equation of the line
N
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: for c
...
f
...
d
...
ranges are 𝑥 ∈ [𝑎, 𝑏)

For continuous RV with p
...
f
...
t
...
2 Normal Distribution (Gaussian)
X: a continuous RV ~ standard normal distribution if p
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f
...
d
...
satisfies:

𝑥

𝐹𝑋 (𝑥) = ℙ(𝑋 ≤ 𝑥) = ∫ 𝑓𝑋 (𝑢) 𝑑𝑢
−∞

Where 𝑓𝑥 (𝑥) satisfies
1
...
∫−∞ 𝑓𝑥 (𝑥) 𝑑𝑥 = 1
𝑑

s
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𝑓𝑋 (𝑥) = 𝑑𝑥 𝐹𝑋 (𝑥) where 𝐹𝑋 (𝑥) is continuous
Thus modelled as 𝑋~𝒩(0, 1); ⁡𝔼(𝑋) = 0, 𝑉𝑎𝑟(𝑋) = 1
2

𝑥
∞ 1
Where ∫−∞
𝑒− 2
√2𝜋

=1

N
...
: 𝔼(𝑋) = 0 as symmetrical about 𝑥 = 0

1
...
(0,
) is a stationary point and a global maximum
√2𝜋

3
...
Transformation of𝑓𝑋 (𝑥) =
a
...
Scaling:

1

1
√2𝜋

𝑒

1
√2𝜋

𝑒

(𝑥−𝑎)2
− 2

𝑥2

−2

valid as integrating from -a to a still gives 1

𝑥2

𝑒 −2𝑎 is not valid as enlarging it changes the area
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Linear transformation:

1
√2𝜋

𝑒

(𝑥−𝑏)2
−
2𝑎

1

not valid
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Φ(−𝑥) = 1 − Φ(𝑥) if 𝑥 > 0 as it is symmetrical about 𝑥 = 0
ℙ(𝑋 ≤ 𝑥) = ℙ(𝜎𝑍 + 𝜇 ≤ 𝑥)
𝑥−𝜇
= ℙ(𝑍 ≤
)
𝜎
𝑥−𝜇
= Φ(
)
𝜎
5
...
The discrete distribution must be approximately normal
2
...
Provided continuity correction is done, where
a
...
5 ≤ 𝑋𝑐 < 𝑏 + 0
...
B
...
ℙ(𝑎 < 𝑋𝑑 ≤ 𝑏) ≈ ℙ(𝑎 + 0
...
5)
c
...
5 ≤ 𝑋𝑐 < 𝑏 − 0
...


MM Applied Cheat Sheet – AS Stats
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𝑋𝑖 are iid RVs (all follow
distribution as the population)
...
Sample mean: 𝑋; = 𝑛 ∑𝑛𝑖=1 𝑋𝑖
1

2
...
Sample variance: 𝑆 2 (𝑜𝑟⁡𝜇2 ); = 𝑛−1 ∑𝑛𝑖=1(𝑋𝑖 − 𝑋) [Sample deviation: √𝑆 2]
Often written 𝑆𝑥𝑥 ; = ∑𝑛𝑖=1(𝑋𝑖 − 𝑋)

2

1

4
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2 Testing Sample Mean Using Normal Distribution
Let 𝑋𝑖 , 𝑖 = 1, 2, … , 𝑛 be a random sample from the population
...


∵ mean less affected by
extreme values
∴ variance less

1
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If 𝜎 2 unknow -> use 𝑆 2 to replace 𝜎 2 provided 𝑛 ≥ 50
1
𝑆 2 = 𝑆𝑥𝑥
𝑛
Where 𝑆𝑥𝑥 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥)2
2
Or = ∑𝑛𝑖=1 𝑥 2 − 𝑛𝑥
Then𝑋~𝒩(𝜇,

𝑆2
𝑛

𝑆

𝑥𝑥
= 𝑛(𝑛−1)
)

Hypothesis Testing
1
...
”
2
...
Layout H0, H1, 𝛼
1
1
4
...
State the 𝜇 under H0 which will be given by the question: “Under H0: 𝑋~(… , … )”
6
...
Compare 𝑥 and 𝑋𝑐
...
If 𝑥 < 𝑋𝑐 : “insufficient evidence to reject H0”
c
...
Conclusion in context

𝑡2
𝜎 2 ( )under
√𝑛

H0:

Let’s say the⁡𝜎 2 in 𝑋𝑖 is 𝑡 2
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1 Terminology of Bivariate Data
- Usually on scatter
- If both var are random and relationship linear, then association is ‘correlation’

8
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3 Measures of Correlation
Pearson’s Product Moment Correlation Coefficient (pmcc)
𝑆𝑥𝑦
𝑟=
√𝑆𝑥𝑥 𝑆𝑦𝑦
2
𝑛
𝑛
2
2
Where 𝑆𝑥𝑥 = ∑𝑖=1(𝑥𝑖 − 𝑥) = ∑𝑖=1 𝑥𝑖 − 𝑛𝑥 ;
2
𝑆𝑦𝑦 = ∑𝑛𝑖=1(𝑦𝑖 − 𝑦)2 = ∑𝑛𝑖=1 𝑦𝑖2 − 𝑛𝑦 ;
𝑆𝑥𝑦 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥)(𝑦𝑖 − 𝑦) = ∑𝑛𝑖=1 𝑥𝑖 𝑦𝑖 − 𝑛𝑥𝑦
...
When data cover whole population, population correlation coefficient denoted by 𝜌
2
...
r is only a sample from a parent bivariate distribution rather than whole population
4
...
3, s
...
n-2:
degrees of freedom denoted by 𝜐 = 𝑠𝑎𝑚𝑝𝑙𝑒⁡𝑠𝑖𝑧𝑒 − 𝑛𝑜

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