Descriptive Stats/Probability

Question 1

What are the conditions or axioms of probability?

Answer 1

1. P(A) ≥ 0
2. P(Ω) = 1
3. For any {A_i} _{i=1 to n} (s.t. the union of A_i & A_j = 0) the P(A_j ∪ A_i) = Σ P(A_i)

Question 2

What can we use Chebyshev’s Inequality for?

Answer 2

For setting the upper bound of a probabilistic event

Question 3

What is Bayes Theorem?

Answer 3

Memorize on paper

Question 4

What is the expected value formula?

Answer 4

E(X) = Σ_i x_i p_x(x_i) **for discrete

E(X) = ∫ x * f_x(x)dx****for continuous

Question 5

What is the total probability theorem?

Answer 5

P(B) = P(B|A) * P(A) + P(B|A^c) * P(A^c)

Question 6

What are the basic operations in probability?

Answer 6

1. Ω = certain event
2. Ø = impossible event
3. If A is an event, then A^c is the complement
4. A ∪ B is true ⇔ A, B, or both are true
5. A ∩ B is true ⇔ A and B are both true
6. A ∩ B = 0 ⇒ A and B are independent (disjoint)

Question 7

What are the properties of the probability measures?

Answer 7

1. If A belongs to B ⇒ P(A) ≤ P(B)
2. P(A) € [0,1]
3. P(A ∪ B) = P(A) + P(B) - P(A ∩ B) (same is true if one element is complement)
4. P(A^c) = 1 - P(A)
5. P(Ø) = 0
6. P(A^c ∩ B) = P(B) - P(A ∩ B)
7. P(A^c ∪ B^c) = 1 - P(A ∩ B) (same is true for intersection)
8. P(A ∩ B) ≥ P(A) + P(B) - 1

Question 8

What are the distributive properties of the probabilistic operations?

Answer 8

1. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
   * The same is true for the union operation*

Question 9

What is the conditional probability of P(A|B)?

Answer 9

P(A ∩ B) / P(B)

***if P(B|A), intersection remains while denominator changes to P(A)

Question 10

What can we say about two random variables, X and Y, if P(X € A) = P(Y € A)?

** or if F_x(z) = F_y(z) ?

Answer 10

They are identically distributed (but not that X = Y)

Question 11

What is Chebyshev’s Inequality?

Answer 11

P( |X - E(X)| ≥ c ) ≤ σ²/c²

**where σ² = Var(X)

Question 12

What is the relationship between the CDF and PDF?

Answer 12

PDF is the derivative of the CDF

Question 13

What is the formula for Variance?

Answer 13

Same as that for mean (both discrete and continuous), but with (x_i - E(X))² substituted for the first x

Question 14

What are the properties of the CDF?

Answer 14

1. The limit as x approaches -∞ equals 0
2. The limit as x approaches ∞ equals 1
3. 0 ≤ F_x(x) ≤ 1
4. F_x(x + h) ≥ F_x(x) **function is not decreasing
5. The limit of Fx(x + h) as h → 0 = F_x(x) **continuous from the right

Question 15

What are the properties of variance?

Answer 15

1. Var(X) = M[(X - μ)²] = M(X²) - μ²
2. Var(X) = 0 ⇔ X = μ (constant)
3. Var(X + a) = Var(X) (just shifts the distribution)
4. Var(bX) = b²Var(X)

Question 16

What is the median?

Answer 16

A specific case of quantile that divides the distribution of a variable into two equal parts (may not be unique if X is discrete)

Question 17

If X is a continuous r.v. and f_x is its density function, what can be said of P(X = x)?

Answer 17

It is 0

**integral of a number from itself to itself is 0

Question 18

What are the properties of the expected value?

Answer 18

1. E(a + bX) = a + bE(X)
2. E [X - E(X)] = 0
3. E [(X - E(X))²] ≤ E [(X - a)²]

Question 19

What are the properties of a PDF?

Answer 19

1. f_x(x) ≥ 0
2. ∫ f_x(x) dx = 1

Question 20

What is the formula for variance?

Answer 20

1. M([X - μ]²)

Question 21

What are the 3 properties of the arithmetic mean?

Answer 21

1. The mean is a linear operator: M(a + bx) = a + bM(x)
2. M(x - μ) = 0
3. μ is the minimizer of M[(x - μ)²]

That is: M[(x - a)²] >= M[(x - μ)²]

Question 22

A and B are stochastically independent if and only if:

Answer 22

P(A ∩ B) = P(A) * P(B)

Question 23

Can the CDF and/or PDF ever be decreasing?

Answer 23

The CDF cannot be

Question 24

What are the properties of Variance?

Answer 24

1. If Var(X) = 0 –> X is constant
2. Var(X) = E(X²) - [E(X)]²
3. Var(a +bX) = b²Var(X)

Question 25

How do we represent the median probabilistically?

Answer 25

P(X ≥ m) ≥ .5 P(X ≤ m) ≥ .5

Question 26

What is Cov(X, Y) if both are discrete?

Answer 26

ΣΣ(x_i - µ_x)(y_j - µ_y) p_x(x_i, y_j) or Σ(X - µ_x)(Y - µ_y) / n

Question 27

What is the formula for Corr(X, Y)?

Answer 27

þ_xy(X, Y) = Cov(X, Y)/σ_xσ_y and -1 ≤ þ_xy(X, Y) ≤ 1 \*\*where þ = correlation coefficient

Question 28

What does it mean if |þ| = 1 ?

Answer 28

The linear association is perfect

Question 29

What are covariance and correlation?

Answer 29

Two indices of linear association between two variables

Question 30

What can be said if Cov(X, Y) = 0 ?

Answer 30

X and Y are statistically independent ## Footnote *\*\*the opposite is NOT necessarily true*