Continuous Probability Distributions: Verifying Probability Density Function

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What is the value of P(0 < X ≤ 1), given the probability density function f(x) = x^2/3, -1 < x < 2?

2/9 (correct)

1/3

1/6

1/9

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 (correct)

F(x) = x^2/3, -1 ≤ x ≤ 2

F(x) = x^2/3 + 1, -1 ≤ x ≤ 2

F(x) = x^3/9 + 1, -1 ≤ x ≤ 2

What is the mean of the random variable X, representing the number of heads in 16 tosses of two coins?

7/16

11/16 (correct)

5

4

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. Signup and view all the answers

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 Signup and view all the answers

What is the value of μ in the given discrete probability distribution?

0.61 Signup and view all the answers

What is the formula for calculating the variance of a continuous random variable X with probability density function f(x)?

σ² = E(X²) - μ² Signup and view all the answers

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) Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

What is the covariance of two random variables X and Y with means μX and μY, respectively?

E[(X - μX)(Y - μY)] Signup and view all the answers

What is the formula for the covariance of discrete random variables X and Y?

∑∑ (x - μX)(y - μY)f(x, y) Signup and view all the answers

What does a negative covariance between X and Y indicate?

A negative correlation between X and Y Signup and view all the answers

What is the covariance of X and Y in the 'ballpoint pens' example?

-56/28 Signup and view all the answers

What is the first step in finding the covariance of X and Y in the 'fraction of runners' example?

Compute the marginal density functions Signup and view all the answers

What is the interpretation of the covariance of X and Y?

The linear relationship between X and Y Signup and view all the answers

What is the term used to describe the long-term average outcome of a given scenario?

Expected value Signup and view all the answers

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 Signup and view all the answers

What is the mean or expected value of X if X is a continuous random variable?

μ = E(X) = ∫xf(x)dx from -∞ to ∞ Signup and view all the answers

What is the expected value of the number of good components in the sample of 3?

1.714 Signup and view all the answers

What is the number of defective components in the lot?

3 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

What is the joint probability distribution of two discrete random variables X and Y?

A joint probability distribution function Signup and view all the answers

What is the relationship between the random variables X and Y in the given problem?

They are independent random variables Signup and view all the answers

What is the value of Z2 = 3X – 2Y + 5 when X and Y are independent random variables with variances σX2 = 2 and σY2 = 3?