Probability Distributions and Investments
37 Questions
1 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What defines a discrete random variable?

  • It can take any real value.
  • It is always associated with a continuous probability distribution.
  • It has an infinite number of possible values.
  • It has a finite number of possible values. (correct)

    • Which function describes the distribution of a continuous random variable?

  • Probability Density Function (correct)
  • Cumulative Distribution Function
  • Event Space Function
  • Probability Mass Function

    • What is the sample space for rolling a six-sided die?

  • {0, 1, 2}
  • {1-36, 0, 00}
  • {H, T}
  • {1, 2, 3, 4, 5, 6} (correct)

    • Which of the following is an example of a random variable mapping outcomes to real values?

    <p>Number of heads in multiple coin flips</p> Signup and view all the answers

    What does the event space represent in probability theory?

    <p>The collection of events related to a random variable</p> Signup and view all the answers

    What is the significance of expressing stock returns as a function of stochastic factors?

    <p>It incorporates uncertainty in outcomes.</p> Signup and view all the answers

    What does the sample space refer to in probability theory?

    <p>The range of possible outcomes for a statistical event.</p> Signup and view all the answers

    Why is it considered impractical to account for all factors affecting stock returns?

    <p>There are too many identifiable factors to consider.</p> Signup and view all the answers

    What does the thought experiment concerning probability focus on?

    <p>Evaluating observed relationships and their randomness.</p> Signup and view all the answers

    In the context of stock price modeling, why is it essential to consider probability?

    <p>It describes uncertainties in data generation.</p> Signup and view all the answers

    What is the correct calculation for P(trade)?

    <p>$0.51$</p> Signup and view all the answers

    What was the main objective of the 'Lady Testing Tea' experiment?

    <p>To verify claims about tea preparation methods.</p> Signup and view all the answers

    How does the concept of probability enhance data analytics?

    <p>It helps estimate the likelihood of events occurring.</p> Signup and view all the answers

    Which condition must be true for Bayes' rule to be applied?

    <p>P(B) &gt; 0</p> Signup and view all the answers

    What does it mean if two events A and B are independent?

    <p>P(A|B) = P(A)</p> Signup and view all the answers

    What type of return model represents the equation: returnit = f(past returnit) + εit?

    <p>Stochastic.</p> Signup and view all the answers

    How is P(Female | Inheritance) calculated based on Bayes' rule?

    <p>P(Inheritance | Female) x P(Female) / P(Inheritance)</p> Signup and view all the answers

    What is the value of P(signal) given the probabilities provided?

    <p>$0.6$</p> Signup and view all the answers

    What is the probability P(trade|no signal)?

    <p>$0.15$</p> Signup and view all the answers

    How can events be classified if knowledge of one event provides no information about the other?

    <p>Independent</p> Signup and view all the answers

    If there are 82 female billionaires out of 650 total billionaires, what is the initial calculation for P(Female)?

    <p>0.126</p> Signup and view all the answers

    What is the expected return (E(r)) of the Cancer Megafund investment?

    <p>11.9%</p> Signup and view all the answers

    What is the standard deviation (SD) of the investment if 150 independent programs are selected?

    <p>34.6%</p> Signup and view all the answers

    Which of the following correctly describes the failure rate in the Exponential Distribution when λ=0.1?

    <p>0.1 failures per year</p> Signup and view all the answers

    What probability distribution is used to calculate the probability of at least 2 hits in a project?

    <p>Binomial Distribution</p> Signup and view all the answers

    What is the annual profit expected from the Cancer Megafund investment if successful?

    <p>$2B</p> Signup and view all the answers

    How is the probability density function (PDF) defined for the Exponential Distribution?

    <p>$λe^{-λx}$</p> Signup and view all the answers

    What does the cumulative distribution function (CDF) of the Exponential Distribution represent?

    <p>The probability of at least one event occurring in an interval</p> Signup and view all the answers

    In a binomial distribution, what does the parameter n represent?

    <p>The number of trials</p> Signup and view all the answers

    What is the correct representation of the number of correct guesses using the hypergeometric distribution?

    <p>X~hypergeometric(N,K,n)</p> Signup and view all the answers

    In a scenario where a lady is tasting tea, how many cups does she randomly pick from the total?

    <p>4 cups</p> Signup and view all the answers

    What type of distribution is used when there is a specified number of draws with replacements?

    <p>Binomial distribution</p> Signup and view all the answers

    Given the events A and B where A is drawing a Jack first and B is drawing a Jack second, what is P(B|A)?

    <p>3/51</p> Signup and view all the answers

    What formula is used to calculate the probability of both events A and B occurring?

    <p>P(A, B) = P(A) * P(B|A)</p> Signup and view all the answers

    In finding P(Coffee | Dark Chocolate), which formula is used?

    <p>P(Dark Chocolate and Coffee) / P(Dark Chocolate)</p> Signup and view all the answers

    If 80% of friends like Dark Chocolate and 20% like both Dark Chocolate and Coffee, what percent of those who like Dark Chocolate also like Coffee?

    <p>25%</p> Signup and view all the answers

    What type of distribution is applicable for draws without replacement?

    <p>Hypergeometric distribution</p> Signup and view all the answers

    Study Notes

    Probability Distributions

    • Binomial Distribution: Describes the number of successes in a fixed number of trials with two possible outcomes (success/failure).
    • Exponential Distribution: Time until an event occurs; characterized by the mean number of events in a time interval (λ).
    • Normal Distribution: Continuous probability distribution symmetric about the mean, depicted by bell-shaped curve.

    The Cancer Megafund

    • Investment Breakdown: $200 million with a 10-year payoff, 5% success probability, and annual profits of $2 billion for 10 years.
    • Expected Return: E(r) = 11.9% with standard deviation SD = 423%.
    • If investing in 150 independent programs: Total investment of $30 billion, E(r) remains at 11.9%, but SD reduces to 34.6%.

    Probability of Hits

    • Question to consider: How to calculate the probability of achieving at least 2 successful outcomes (hits).
    • CDF and PDF Relation: The conversion from Probability Density Function (PDF) to Cumulative Distribution Function (CDF) is notable for understanding event probabilities.

    Conditional Probability

    • Explaining Relationships: Conditional probability aids in linking events where the occurrence of one event affects the probability of another event.
    • Example in Card Drawing: P(A) = probability of drawing a Jack on the first draw, and P(B|A) = probability of a Jack on the second draw given the first was a Jack.

    Lady Tasting Tea

    • Historical Context: In the 1920s, an experiment determined whether a lady could correctly identify the order of milk and tea poured.
    • Sample Space Analysis: Possible outcomes include all combinations of milk and tea presentation.

    Law of Total Probability (LTP)

    • Fundamental Principle: Given two events, it states how to calculate the overall probability considering the partitioning of the sample space.

    Bayes’ Rule

    • Application: Allows the reverse calculation of conditional probabilities, allowing you to determine P(A|B) given P(B|A).
    • Example with Inherited Wealth: Used in assessing the proportions of billionaires who inherited their fortunes based on gender.

    Independence

    • Concept Definition: Events A and B are independent if the occurrence of A does not influence the occurrence of B, mathematically expressed as P(A|B) = P(A) and P(B|A) = P(B).

    Random Variables

    • Definition: A function that maps outcomes of a random process to numerical values.
    • Types of Random Variables:
      • Discrete: Finite set of outcomes (e.g., results of a dice roll).
      • Continuous: Infinite set of outcomes (e.g., asset prices).

    Key Distributions

    • PMF (Probability Mass Function): Used for discrete random variables indicating the probability of obtaining specific values.
    • PDF (Probability Density Function): Pertains to continuous random variables, representing the probability distribution of the variable.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Probability, Random Variables, and Distributions (BMF5324) PDF

    Description

    This quiz covers key concepts in probability distributions, including binomial, exponential, and normal distributions. Additionally, it explores investment breakdowns and expected returns in financial contexts, particularly relating to the Cancer Megafund. Test your understanding of these statistical principles and their applications in real-life scenarios.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser