Discrete probability distributions are used for variables that take on countable values. The price of a house is a continuous variable because it can take any value within a range (e.g., $250,000.50). Conversely, the number of bedrooms and the presence of a pool are discrete, as they represent distinct, countable categories or integers.
142
A continuous random variable is defined as one that can assume any value within a specific _______________.
A continuous random variable is characterized by its ability to take on any real value within a defined range or interval. Unlike discrete variables, which take on countable values, continuous variables exist on a continuum where there are infinitely many possible values between any two points.
143
What term describes a comprehensive list of all possible outcomes of an experiment paired with their respective probabilities?
A probability distribution is a mathematical function or table that provides the probabilities of occurrence of different possible outcomes for an experiment. It fully describes the random phenomenon by assigning a probability to each value in the sample space, ensuring that the sum of all probabilities equals one for discrete variables.
144
For a cumulative distribution function F(x), given that F(-∞) = 0, what is the value of F(∞)?
The cumulative distribution function F(x) is defined as P(X ≤ x). By the axioms of probability, as x approaches positive infinity, the probability of the event X ≤ x approaches 1. Therefore, F(∞) must equal 1. Since option C is 1, the provided answer D is factually incorrect.
145
Which distribution utilizes the formula mn / ((m + n)^2 * (m + n + 1)) to calculate its variance?
The variance of a Beta distribution with parameters m and n (often denoted as α and β) is given by the formula [αβ] / [(α+β)^2 * (α+β+1)]. This formula describes the spread of the distribution based on its shape parameters.
146
For a uniform distribution defined on the interval [a, b] where a = 2.5 and b = 3.5, what is the mean of the distribution?
The mean of a continuous uniform distribution on the interval [a, b] is calculated as (a + b) / 2. Substituting the given values: (2.5 + 3.5) / 2 = 6 / 2 = 3. This represents the central point of the uniform distribution.
147
Which term is commonly used to describe the cumulative distribution function of a random variable?
In statistical literature, the cumulative distribution function (CDF) is frequently referred to simply as the 'distribution function'. It uniquely defines the probability distribution of a random variable by specifying the probability that the variable is less than or equal to any given value. While 'probability distribution' is a broader term, 'distribution function' specifically identifies the CDF.
148
What is the fundamental property of a probability function regarding its range of values?
By the axioms of probability, the probability of any event must be a value between 0 and 1 inclusive. Therefore, a probability mass function or probability density function must always be non-negative. It cannot take negative values, as negative probability is undefined in standard Kolmogorov axioms.
149
What is the theoretical mean deviation of a normal distribution in terms of its standard deviation (σ)?
The mean deviation of a normal distribution is approximately 0.7979σ. The provided answer '45σ' is mathematically incorrect as the coefficient should be approximately 0.8. This source answer appears to be a typo or error in the provided dataset.
150
For a normal distribution with a mean of 65, a standard deviation of 25, and a value (x) of 35, what is the standardized random variable (z-score)?
The z-score is calculated using the formula z = (x - μ) / σ. Substituting the given values: z = (35 - 65) / 25 = -30 / 25 = -1.2. This standardized value indicates how many standard deviations the data point lies from the mean.
