29 Questions
What is the value of P(0 < X ≤ 1), given the probability density function f(x) = x^2/3, -1 < x < 2?
2/9
What is the cumulative distribution function F(x) of the continuous random variable X with density function f(x) = x^2/3, -1 < x < 2?
F(x) = x^3/9, -1 ≤ x ≤ 2
What is the mean of the random variable X, representing the number of heads in 16 tosses of two coins?
11/16
What is the nature of the probability density function of a continuous random variable?
It is a non-negative function that integrates to 1 over the entire real line.
What is the purpose of the cumulative distribution function F(x) of a continuous random variable X?
To calculate the probability of a range of values of X
What is the value of μ in the given discrete probability distribution?
0.61
What is the formula for calculating the variance of a continuous random variable X with probability density function f(x)?
σ² = E(X²) - μ²
What is the formula for calculating the variance of a function g(X) of a discrete random variable X?
σ² = ∑[g(X) - μg(X)]²f(x)
What is the value of E(X) for the continuous random variable X with probability density function f(x) = 2(x-1), 1 < x < 2?
4/3
What is the formula for calculating the variance of a function g(X) of a continuous random variable X?
σ² = ∫[g(X) - μg(X)]²f(x)dx
What is the value of σ² for the continuous random variable X with probability density function f(x) = 2(x-1), 1 < x < 2?
5/18
What is the covariance of two random variables X and Y with means μX and μY, respectively?
E[(X - μX)(Y - μY)]
What is the formula for the covariance of discrete random variables X and Y?
∑∑ (x - μX)(y - μY)f(x, y)
What does a negative covariance between X and Y indicate?
A negative correlation between X and Y
What is the covariance of X and Y in the 'ballpoint pens' example?
-56/28
What is the first step in finding the covariance of X and Y in the 'fraction of runners' example?
Compute the marginal density functions
What is the interpretation of the covariance of X and Y?
The linear relationship between X and Y
What is the term used to describe the long-term average outcome of a given scenario?
Expected value
What is the probability distribution of X in the given scenario, where X is the number of good components in the sample?
f(x) = Cx × 3 / 7C3
What is the mean or expected value of X if X is a continuous random variable?
μ = E(X) = ∫xf(x)dx from -∞ to ∞
What is the expected value of the number of good components in the sample of 3?
1.714
What is the number of defective components in the lot?
3
If 3 components are selected at random from a lot of 4 good components and 3 defective components, what is the average number of bad components that one will pick?
1.3
What is the variance of the random variable Z = 3X – 2Y + 5 when X and Y are independent random variables with variances σX2 = 2 and σY2 = 3?
30
What is the joint probability distribution of two discrete random variables X and Y?
A joint probability distribution function
What is the relationship between the random variables X and Y in the given problem?
They are independent random variables
What is the value of Z2 = 3X – 2Y + 5 when X and Y are independent random variables with variances σX2 = 2 and σY2 = 3?
30
What is the purpose of the moment-generating function in joint probability distributions?
To find the moment-generating function of the joint distribution
What is the chapter that discusses the joint probability distribution of two random variables?
Chapter 5
Verify whether a given probability density function integrates to 1 and find the probability of a certain range of values. This quiz deals with continuous probability distributions and probability density functions.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
