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Econometrics > 1 Probability and random variable > Flashcards

1 Probability and random variable Flashcards

1
Q

What is an experiment?

A

Procedure that can be repeated many times and has a well defines set of outcomes.

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2
Q

How do you write the probability for the event A?

A

P(A)

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3
Q

What does means “P(A)”?

A

Proportion of times the event A will occur in repeated trials of an experiment.

For a large number of trials, a relative frequence will provide a good approximation of the probability of A.

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4
Q

What is a “function”?

A

Relation between a set of input & a set of possible outputs, with the property that each input is related to exactly ONE output.

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5
Q

What are the properties for a P(A), a real-valued function?

Function defigned in R.

A

0≤P(A)≤1

If A, B, C ….. constitute an exhaustive set of events, P(A+B+C+…) = 1 where A+B+C means A or B, or C and so forth

If A, B, C … are mutually exclusive events,

P(A+B+C+…) = P(A)+P(B)+P(C)+…

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6
Q

What is a random variable?

A

Numerical variable whose value is determined by the outcome of a random experience.

This value is unknown until observed.

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7
Q

What type can be a random variable?

A

Discrete: take only a finite number of values

Continuous: it ca take on any value in some interval of values & take on any particular value with a zero probability, because there is so many possibilities. Each one has a probility of 0 to happen, statistically.

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8
Q

How is a random variable denoted?

A

X, Y, Z…

Its values are {x, y, z…}

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9
Q

What does indicate a discrete probability density function?

A

P(X=xi) indicates the probability that the discrete random variable X takes the value xi.

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10
Q

How do you write the discrete PDF?

A
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11
Q

For a continuous function, what is the propability for a specific value?

A

0

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12
Q

For a continuous variable, how do you determine the probability of an event?

A

You must take an interval and calculate the probability of getting this event as the outcome.

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13
Q

How do you write the continuous PDF?

A

Where P(a

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14
Q

What is an integral for a continuous PDF?

A

The area under the PDF between the points a and b.

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15
Q

What is a cumulative distribution function (CDF)?

A

It is a sum of all the probabilities between the minimum and xi.

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16
Q

What is the formula of the CDF?

A
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17
Q

Properties of the CDF, for all important continuous distributions:

  1. P(X>c) = ?
  2. P(X>c) = ?
  3. P(X< -c) = ?
  4. For any a < b, P(a < X ≤ b) = ?
A
  1. P(X>c) = P(X≥c)
  2. P(X>c) = 1-F(c)
  3. P(X< -c) = P(X>c) if symmetric
  4. For any a < b, P(a < X ≤ b) = F(b) - F(a)
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18
Q

What is a Discrete Joint PDF?

A

Probability observ outcome x of X and y of Y at the same time.

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19
Q

How do you compute teh Discrete Marginal PDF?

A
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20
Q

What is a conditional PDF?

A

Probability that X takes the value x given that Y has assumed the value y.

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21
Q

What is the formula for a Conditional PDF?

A
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22
Q

When are 2 random variables statistically independent?

A

If the joint PDF can be expressed as the product of the marginal PDFs for all combinations of X and Y.

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23
Q

If X and Y are independent, then, f(x⎪y) = ?

A

f(x⎪y) = f(x)

y doesn’t convey any information on x’s distribution.

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24
Q

WHat is an expected value?

A

It is the (population) mean of the distribution.

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25
Q

What is the formula for the expected value for a discrete distribution?

A
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26
Q

What is the formlula of the expected value for a continuous distribution?

A
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27
Q

Properties of the expected value:

  1. If a is constant, E(a) = ?
  2. if a & b are constant, E(aX + b) = ?
  3. For 2 rv X &Y, E(X, Y) = ?
  4. For a function g(.), E[g(x)] = ?
  5. If X & Y are 2 independent variables, E(XY) = ?
A
  1. If a is constant, E(a) =a
  2. if a & b are constant, E(aX + b) = aE(X) + b
  3. For 2 rv X &Y, E(X, Y) = E(X) + E(Y)
  4. For a function g(.), E[g(x)] = Σxg(x)f(x)
  5. If X & Y are 2 indep var, E(XY) = E(X)E(Y)
28
Q

What is a Median m?

A

It measures the central tendency of a distribution.

  • Continuous: value such that 1/2 of the area under the PDF is set at the left of m and 1/2 is at the right of m.
  • Discrete
    • X takes on a finite numer of odd values: order the values of X and select the one in the middle.
    • X takes on a even number of values: average the 2 middle ones.
29
Q

If the distribution is not symmetric around the mean, what happen to the expected value and the median?

A

The differ.

30
Q

What is the variance?

A

It is the expected distance from the mean.

31
Q

What is the formula of the variance?

A

E(X)=μ

var(X) = E(X)²- [E(X)]²

Or

32
Q

What are the other formulas of the variance?

A
33
Q

Is E[g(x)] = g[E(x)]?

A

NO!!!!!!

It is a non-linear function.

34
Q

Properties of the variance:

A
35
Q

What is covariance?

A

It measures the amount of linear dependence between 2 rvs.

36
Q

What is the formula for covariance?

A
37
Q

If X & Y are discrete, what is the formula for the covariance?

A
38
Q

Properties of covariance?

A
39
Q

What is a correlation?

What is its particularty compared to covariance?

A

Correlation (ρ) is a measure of the linear association between 2 variables.

The correlation doesn’t depend on the unit of measurement, whereas covariance does.

40
Q

What are the max (& min) values of ρ?

What are their meanings?

A

+1 and -1

  • +1: perfect positive association.
  • -1: perfect negative association.
41
Q

What is the formula of the correlation?

A
42
Q

Variance of correlated variables:

var(X +Y)= ?

var(X −Y)= ?

A

var (X+Y) = var(X)+var(Y)+2cov(X,Y)

var(X −Y)=var(X)+var(Y)−2cov(X,Y)

If independence, the thirs term drops.

43
Q

Why do we use the Z transformation?

A

It indicates the number of time the observation is far away from the mean in σ.

44
Q

How do you compute a Z score?

A

Z = (x-μ)/σx

45
Q

σ²z = ?

A

σ²z = 1

46
Q

What is the Conditional expectation of Y given X?

A

It is a weighted average of possible values of Y, but the weights reflect the fact tha X has taken on a specific value.

47
Q

What is the PDF of a rv said to be normally distributed?

A
48
Q

How is denoted a normally distributed rv?

A

X ∼N(μ, σ²)

49
Q

What are the 2 properties of a normally distributed rv?

A
  • It is symmetrical around the mean
  • It only deepends on μ and σ²
50
Q

How is represented a normally distributed rv?

A
51
Q

How is represented the CDF of a normallt distributed rv?

A
52
Q

What are the mean and the unit for a Z score?

A

Mean: 0

Unit: σ

53
Q

What is the central limit theorem?

A

This kind of distribution is always normal.

54
Q

What is the Chi2

A

It is a distribution based on the sum of random variables distributed normally.

Z posses the Chi2 distribution with k degree of freedom (df).

55
Q

What is a Student’s distribution?

A
56
Q

What is inference?

A

It is a process that allows us to learn something about a population given the availability of a random sample from that population.

57
Q

What are the steps of the inference?

A
  1. Identify a relevant population
  2. Draw a sample
  3. Specify a model
  4. Estimate (point or interval) & hypothesis testing
58
Q

What is a random sample?

A

Subset of individuals chosen from a population such that:

  • Each individual is chosen randomly and entierly by chance.
  • Each subset of k individuals has the same chance to be chosen.
  • It can be done with or without replacement (n → ∞).
59
Q

What are the main analyses we can compute on a sample?

A
  • Mean / sample average
  • Sample variance (that follows a Chi2 distribution).
  • Sample covariance.
60
Q

What is an estimator?

A

Rule assigning to an unknown parameter of the underlying population distribution a unique value for each possible sample realization.

We can have plenty of different estimators for the same unknown paramter, each one is a rv.

61
Q

How do you choose the right estimator?

A

Look at the sampling distribution.

62
Q

When is an indicator unbiased?

A

When: E(W)=θ

Maybe the estimator is far from the parameter, but if we do infinite times, it is fine.

63
Q

What is a confidence interval?

A

Computing the smallest interval possible in which θ has the major probability to be.

θ is the value for the population, so we don’t know it.

64
Q

What happen to a confidence interval when then sample is bigger and bigger?

A

It becomes smaller and smaller.

65
Q

When is an estimator consistent?

A

When the distribution of this estimator becomes more and more concentrated near the true value of the paramter being estimated.

Econometrics (2 decks)

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