Econ 10003: Random Variables

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Questions and Answers

Mapping from the sample space to the real line is a characteristic of what?

  • Descriptive statistics
  • A random variable (correct)
  • Inferential statistics
  • A continuous function

Which of the following is NOT a key feature of a discrete random variable?

  • Has a countable number of distinct values.
  • Has a finite number of distinct values.
  • Values are on a subset of the real line.
  • Can take on any value within a given range. (correct)

What does the probability mass function (PMF) of a random variable X represent?

  • The probability that X is less than or equal to a certain value.
  • The probability that X is greater than a certain value.
  • The probability that X is equal to a specific value. (correct)
  • The average value of X.

If a random variable X can only take two values, 0 and 1, what type of random variable is it?

<p>Bernoulli Random Variable (C)</p>
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An indicator random variable X is defined as X=1 if event A occurs and X=0 otherwise. What must be true about events A and its complement?

<p>They must be mutually exclusive and exhaustive. (D)</p>
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With a single throw of a fair six-sided die, what is the probability of getting a number greater than 4 according to its Probability Mass Function?

<p>$1/3$ (B)</p>
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What is a key property of a Cumulative Distribution Function?

<p>Increasing (C)</p>
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What does the expectation of a random variable represent?

<p>A measure of central tendency. (A)</p>
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How is the expected value of a discrete random variable calculated?

<p>By summing the product of each possible value and its probability. (D)</p>
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For a Bernoulli random variable, if 'p' represents the probability of success, what is its expected value?

<p>p (D)</p>
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An insurance company models payouts for a natural disaster. They estimate the following probabilities: No disaster (probability 0.6, payout 0), Storm (probability 0.3, payout 600 pesos), Storm and Flood (probability 0.1, payout 1000 pesos). What is the expected payout?

<p>280 pesos (D)</p>
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What does the variance of a random variable measure?

<p>The spread or dispersion of the data around the mean. (B)</p>
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How can the variance of a random variable X be computed?

<p>$E[X^2] - (E[X])^2$ (D)</p>
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What is the variance of a Bernoulli distribution with success probability 'p'?

<p>p(1 - p) (D)</p>
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Which of the following is the MOST direct application of a random variable?

<p>Mapping sample spaces to numerical values (D)</p>
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In the context of random variables, what does 'countable' refer to when describing the number of distinct values?

<p>Values that can be listed or enumerated, even infinitely. (C)</p>
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Why is assigning numbers to elements in the Sample Space S necessary for statistical analysis?

<p>It forms the basis for quantitative analysis and modeling. (C)</p>
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Given the PMF for a fair six-sided die, what is the probability of rolling an even number?

<p>$1/2$ (C)</p>
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Which of the following is the characteristic of the Cumulative Distribution Function (CDF)?

<p>It provides probabilities for intervals. (D)</p>
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Which of the following statements best describes the role of expectation in decision-making under uncertainty?

<p>It provides a probability-weighted average of outcomes to help guide choices. (B)</p>
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A random variable X is defined such that X = 1 with probability p and X = 0 with probability 1-p. What is $E[X^2]$?

<p>p (C)</p>
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How does climate change potentially destabilize the Insurance Industry, according to the information provided?

<p>By causing more frequent and severe natural disasters. (A)</p>
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A life insurance company wants to model the economic risks of a certain disease. Why would they use a random variable to model this?

<p>To create a theoretical model for analysis, due to the inherent unpredictability. (D)</p>
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What key challenge do climate change and natural disasters pose to the insurance industry, as hinted by the insurance problem example?

<p>Difficulty in creating accurate payout models (A)</p>
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If for every 1000 pesos of insured property, damaged inventories result in a payout of 600 pesos for a storm and 1000 pesos for a combined storm and flood, what does this describe in the context of insurance modeling?

<p>Contingent liabilities (A)</p>
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What does the assumption "climate change does not worsen" imply for the insurance problem?

<p>The model can use historical payout data with more reliability (A)</p>
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In the context of an insurance payout as a random variable, what does a higher payout variance suggests to the insurance company?

<p>Higher financial risk (C)</p>
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Consider a simplified probability model where an insurance company in the Philippines distinguishes three events on a yearly basis: no natural disaster, a storm and a storm and flood. If they want to estimate their potential yearly losses, which tool should they use?

<p>The expected value of the payouts (C)</p>
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Which of the following real-world phenomena can NOT be appropriately modeled by a random variable?

<p>The outcome of a deterministic physics equation. (D)</p>
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Wage (W) and Education (E) are discrete random variables. Given that $P(W=High, E=College) = 0.19$, $P(W=High, E=Not College) = 0.12$, $P(W=Low, E=College) = 0.17$ and $P(W=Low, E= Not College) = 0.52$, what is the probability of having a high wage?

<p>0.31 (D)</p>
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Wage (W) and Education (E) are discrete random variables. Given that $P(W=High, E=College) = 0.19$, $P(W=High, E=Not College) = 0.12$, $P(W=Low, E=College) = 0.17$ and $P(W=Low, E= Not College) = 0.52$, what is the probability of having a college education?

<p>0.36 (C)</p>
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To calculate Variance in an insurance problem, what must the company first estimate?

<p>The expected value of a disaster (D)</p>
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Suppose $f(x) = 1/3$, for $x = -1, 0, 1$, what is $E[x]$?

<p>0 (A)</p>
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Suppose $f(x) = 1/3$, for $x = -1, 0, 1$, what is the variance?

<p>2/3 (C)</p>
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An economist is modeling the annual income of individuals in a city. He decides to use a random variable. What is the most important assumption he is making?

<p>The annual income has some element of randomness. (A)</p>
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Why is statistical analysis difficult to use with climate change disasters and insurance?

<p>Because climate change brings unpredictability (D)</p>
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In probability, why is a random sample important?

<p>Each member of the population has an equal chance fo being selected (B)</p>
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Flashcards

What is a random variable?

Mapping from the sample space to the real line.

What is a discrete random variable?

A random variable which takes only discrete values.

What is Probability Mass Function (PMF)?

The probability that a discrete random variable X takes a specific value.

What is a Bernoulli Random Variable?

A random variable with only two possible values: 0 and 1.

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What happens if X(success)=1?

The experiment results in success.

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What happens if X(failure)=0?

The experiment results in failure.

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What is an Indicator Random Variable?

A random variable with two values that are mutually exclusive and exhaustive.

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What is the Expectation of a Random Variable?

A measure of central tendency that shows the average value.

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What happens if X=1 with probability p?

The average value of variable X is 1.

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What happens if X=0 with probability 1-p?

The average value of variable X is 0.

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Climate and the Insurance Industry?

Climate

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Study Notes

  • Probability, Statistics and Econometrics Econ 10003
  • Random Variables - 2025

Random Variables

  • Mapping from the sample space to the real line
  • Used in games of chance, like coin tosses or single dice throws
  • Assigns a number corresponding to an element in the Sample Space S
  • Forms the basis for statistical analysis
  • Following a single flip of a coin, a random variable X can be defined
  • The values X will take before the coin flip are unknown

Discrete Random Variable

  • Discrete: X only takes values on a subset of the real line
  • Countable number of distinct values; finite number of distinct values
  • Probability mass function (PMF) of a random variable X.
  • PMF notations include f
  • Example:
    • X = 1, P(X=x) = 0.5 (from a single coin toss)
    • X = 0, P(X=x) = 0.5 (from a single coin toss)

Joint Probabilities

  • Wage and Education
  • H = High Wage
  • L = Low Wage
  • C = College
  • N = No College
  • Education E, P(E=e)
    • C = 1 = 0.36
    • N = 0 = 0.64
  • Wage W, P(W=w)
    • H = 1 = 0.31
    • L = 0 = 0.69

Wages earners in US (2009)

  • The distribution of wages in the US in 2009
    • W = 1 = 0.3
    • W = 0 = 0.7

College attendance in the US (2009)

  • The distribution of college attendance in the US in 2009
    • C = 1 = 0.36
    • C = 0 = 0.64

Bernoulli Random Variable

  • Random variable X has only two possible values: 0 and 1
  • X(success) = 1 in a random experiment
  • X(failure) = 0 in a random experiment
  • P(X=1)=p
  • P(X=0)=1-p
  • The PMF of X can be written as

Indicator Random Variable

  • Random variable X has only two possible values which are mutually exclusive and exhaustive
  • X=1 if A occurs (e.g. an individual is a high wage earner)
  • X=0 otherwise (e.g. an individual is a low wage earner)
  • P(X=1)=p
  • P(X=0)=1-p

Probability Mass Function

  • The PMF for years of education in US (gathered from US data)
    • X = Years of Education (US Wage Earners in 2009)
      • 8 = 0.027
      • 9 = 0.011
      • 10 = 0.011
      • 11 = 0.026
      • 12 = 0.274
      • 13 = 0.182
      • 14 = 0.111
      • 16 = 0.229
      • 18 = 0.092
      • 20 = 0.037

Probability Mass Function

  • PMF from a single throw of a dice is 1/6

Cumulative Distribution Function

  • Cumulative Distribution Function plot
  • Cumulative Distribution Function properties
  • Increasing
  • Right continuous
  • Cumulative Distribution Function (Find probabilities for intervals)

Expectation of a random variable

  • Measure of central tendency
  • Denoted as E(X)
  • Average value with probability-weighted averaging
  • Expected value, average or mean of the distribution
  • Where denotes the PMF

Expectation of a Random Variable

  • X=1 with probability p and X=0 with probability 1-p which yields and expected value
  • For a Bernoulli random variable, same as the probability of a success
  • Examples include the US wage, college attendance
  • The average number of years of education is about 14

Insurance Problem

  • Climate change is Destabilizing Insurance Industry, scientific American in 2003
  • Every year 26 million people are pushed into poverty by natural disasters (The World Bank, 2017)
  • Every year disasters cause an average $300 billion in economic losses (The World Bank, 2017)

Applications of Expectation

  • An insurance company based in the Philippines decided to use a simplified probability model to describe the risks of natural disasters due to climate change
  • Yearly Basis:
    • No natural disaster with a probability of 0.6
    • A storm disaster occurring with a probability of 0.3
    • A combination of storm and flood disaster occurring with a probability of 0.1
  • For every 1000 pesos of insured property, damaged inventories of recent years tell the following
    • Payout is 600 pesos for a storm
    • Payout is 1000 pesos for a combined storm and flood
    • Zero payout when no disaster happens
  • X is a random variable that represents the payout
  • There is a likelihood of a storm occurring in 30% of the years, combined storm and flood in 10% of the years and no disaster in 60% of the years
  • Assuming climate change does not worsen, and property insurance has a long duration
  • The average yearly payout

Variance

  • Variance of X
  • Insurance problem
  • Standard deviation
  • Variance of X: can be computed as
  • Let X be a random variable with the PMF, computing the mean and variance
  • Let Y be a random variable with the PMF g, computing the mean and variance

Variance of a Bernoulli Distribution

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