Chapter 4 Flashcards
What is a continuous random variable?
A variable that can take any value within a given range.
What is the key difference between discrete and continuous random variables?
Discrete random variables have countable outcomes, while continuous random variables have uncountable outcomes.
Why does P(X = x) = 0 for continuous random variables?
Because there are infinitely many possible values, making the probability of any single value zero.
What is the cumulative distribution function (CDF)?
A function that gives the probability that a random variable is less than or equal to a given value.
What is the probability density function (PDF)?
A function that describes the likelihood of a continuous random variable taking on a specific value.
What is the relationship between the CDF and PDF?
The PDF is the derivative of the CDF.
What condition must a PDF satisfy?
It must be non-negative and integrate to 1 over its entire range.
How do you compute the probability of a range of values for a continuous random variable?
By integrating the PDF over that range.
What is the expected value of a continuous random variable?
The integral of x times the PDF over all possible values.
What is the formula for expectation in continuous variables?
E(X) = ∫ x fX(x) dx.
What is variance in continuous probability distributions?
A measure of how spread out the values of a random variable are.
What is the formula for variance?
Var(X) = E(X²) - (E(X))².
What is the uniform distribution?
A distribution where all values in an interval are equally likely.
What is the PDF of the uniform distribution?
fX(x) = 1 / (b - a) for a ≤ x ≤ b, otherwise 0.
What is the expected value of a uniform distribution?
E(X) = (a + b) / 2.
What is the variance of a uniform distribution?
Var(X) = (b - a)² / 12.
What is the exponential distribution used for?
Modeling the time between independent events.
What is the PDF of the exponential distribution?
fX(x) = λ e^(-λx) for x ≥ 0, otherwise 0.
What is the expected value of an exponential distribution?
E(X) = 1 / λ.
What is the variance of an exponential distribution?
Var(X) = 1 / λ².
What is the memoryless property of the exponential distribution?
P(X > x + t | X > t) = P(X > x), meaning past waiting time does not affect future probabilities.
What is the normal (Gaussian) distribution?
A continuous probability distribution that is symmetric around its mean.
What is the PDF of a normal distribution?
fX(x) = (1 / (σ√(2π))) e^(-(x - μ)² / (2σ²)).
What are the parameters of a normal distribution?
Mean (μ) and standard deviation (σ).
What is the standard normal distribution?
A normal distribution with mean 0 and variance 1.
What is the CDF of the normal distribution called?
The standard normal cumulative distribution function (Φ).
Why is the normal distribution important?
Because of the Central Limit Theorem, which states that sums of many independent random variables tend to follow a normal distribution.
What is a linear transformation of a normal variable?
If X ~ N(μ, σ²), then Y = aX + b is also normally distributed.
How is the probability of a normal variable computed?
Using the standard normal table or numerical integration.
What is the key takeaway about continuous distributions?
They are described by PDFs and CDFs, and probabilities are computed via integration.
