A discrete probability distribution assigns probabilities to specific, countable outcomes. It defines the likelihood of each individual outcome in a discrete set, such as rolling a die or drawing a card from a deck.
What does a topology in mathematics primarily deal with?
A Shapes and spaces
B Structure of groups
C Probability theory
D Linear equations
Topology is the branch of mathematics that deals with the properties of space that are preserved under continuous deformations like stretching or bending. It focuses on concepts like continuity, compactness, and convergence.
What is the main objective of decision theory?
A Finding the shortest path
B Making optimal choices in uncertain situations
C Analyzing graph structures
D Solving mathematical proofs
Decision theory is concerned with making rational choices in situations where the outcome is uncertain. It uses mathematical models to evaluate different strategies based on probabilities, costs, and benefits.
In a discrete probability distribution, what does the sum of all probabilities equal?
A 1
B 0
C The number of outcomes
D 100%
In a discrete probability distribution, the sum of all probabilities for all possible outcomes must equal 1. This ensures that one of the possible outcomes must occur, reflecting certainty in the probability model.
What does the principle of continuity in topology deal with?
A Pathway between points
B Discrete sets
C Graph cycles
D Optimal paths
In topology, continuity refers to the idea that a function between two spaces allows for a smooth transition without breaks or jumps. It means that small changes in input result in small changes in output.
What is a random variable in discrete probability distributions?
A A fixed outcome
B A function that assigns probabilities
C A value that depends on the outcome of a random experiment
D A constant
A random variable is a variable whose value is determined by the outcome of a random experiment. It assigns a numerical value to each possible outcome, and can be discrete or continuous.
In decision theory, what is a payoff matrix used for?
A To calculate probabilities
B To find optimal strategies for different players
C To model graph algorithms
D To find the shortest path
A payoff matrix is used in decision theory to represent the outcomes (payoffs) for each combination of strategies by different players. It helps in determining optimal strategies in games or decision making scenarios.
What is a discrete random variable?
A A variable with an infinite set of outcomes
B A variable that takes on distinct, countable values
C A variable defined by a continuous distribution
D A constant variable
A discrete random variable is one that can take on a finite or countable number of distinct outcomes. Examples include the number of heads in coin flips or the number of people in a room.
Which of the following is a common application of decision theory?
A Traffic flow analysis
B Making investment choices
C Calculating binomial coefficients
D Solving algebraic equations
Decision theory is often applied in areas like finance and economics to make optimal investment choices under uncertainty. It helps in evaluating the potential outcomes of different decision making strategies.
What is a characteristic function in probability theory?
A A function that defines the probability of events
B A function that transforms random variables
C A function that describes a distribution’s moment
D A function that maps events to outcomes
A characteristic function in probability theory is used to describe the distribution of a random variable. It provides a way to analyze the properties of distributions and is defined as the expected value of the complex exponential of the random variable.
What is the Hausdorff dimension in topology?
A A measure of the “size” of a space
B The dimension of Euclidean space
C A concept in algebraic topology
D A measurement of graph edges
The Hausdorff dimension is a measure in topology that generalizes the concept of the dimension of a space, particularly useful in fractals and non integer dimensional spaces, capturing the complexity of structures like coastlines.
What does the concept of “compactness” in topology refer to?
A The smallest possible set
B A set that is closed and bounded
C The largest subset
D A path with no breaks
In topology, a set is compact if it is closed (contains all its boundary points) and bounded (can be enclosed in a finite region). Compactness is a key concept in analysis and functional spaces.
What is the probability of getting exactly 2 heads when flipping 3 coins?
A 1/8
B 3/8
C 1/4
D 1/2
The probability of getting exactly 2 heads when flipping 3 coins can be calculated using the binomial distribution. There are 3 possible favorable outcomes (HHT, HTH, THH), out of 8 total possible outcomes, so the probability is 3/8.
In decision theory, what is a dominant strategy?
A A strategy that is always the best, regardless of the opponent’s choice
B A strategy that depends on the opponent’s choice
C A strategy that minimizes losses
D A strategy with the highest payoff
A dominant strategy in decision theory is one that always results in the highest payoff for a player, regardless of the strategies chosen by the other players. It is a key concept in non cooperative games.
What is the formula for the variance of a discrete random variable?
A Sum of probabilities
B Average of outcomes
C Sum of squared deviations from the mean
D Average probability of outcomes
The variance of a discrete random variable is the average of the squared differences between each outcome and the mean. It measures the spread or dispersion of the values around the mean.