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Quiz 8: Continuous Random Variables

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Quiz

This quiz tests materials in the section called “Continuous random variables”. If you are struggling with the questions go back and re-read the relevant material.

Suppose that a continuous random variable \(X\) has a probability density function of the form:

\[ f(x) = \frac{x^3}{10} - \frac{x}{20} + \frac{7}{20} \qquad 0 \leq x \leq 2; \ \ \ 0 \mbox{ otherwise} \]

Question 1: Calculate the cumulative density function, \(F(x)\)

Choose the correct answer:

  1. \(F(x) = \frac{x^2}{20} - \frac{1}{20} - \frac{7x^{-1}}{20}\)

  2. \(F(x) = \frac{4 x^4}{10} - \frac{2 x^2}{20} + \frac{7x}{20}\)

  3. \(F(x) = \frac{x^4}{40} - \frac{x^2}{40} + \frac{7x}{20}\)

  4. \(F(x) = \frac{3 x^2}{10} - \frac{1}{20}\)

[Hint: as a check, what value should F(0) and F(2) take?]

Question 2: Calculate \(E(X)\)

Choose the correct answer:

  1. \(0.344\) (3 d.p.)

  2. \(1.8\)

  3. \(1.207\) (3 d.p.)

  4. \(1.52\)

Question 3: Calculate \(E(X^2)\)

Choose the correct answer:

  1. \(0.344\) (3 d.p.)

  2. \(1.8\)

  3. \(1.207\) (3 d.p.)

  4. \(1.52\)

Question 4: Calculate \(Var(X)\).

Choose the correct answer:

  1. \(0.344\) (3 d.p.)

  2. \(1.8\)

  3. \(1.207\) (3 d.p.)

  4. \(1.52\)

The random variables \(X\) and \(Y\) have joint density function

\[ f(x, y) = 12xy(1 - x),\qquad \mbox{if } 0 < x < 1 \mbox{ and } 0 < y < 1; \ \ \ 0 \mbox{ otherwise} \]

Question 5: What is the marginal density of \(Y\)?

Choose the correct answer:

  1. \(2y\)

  2. \(6y(1-y)\)

  3. \(6x(1-x)\)

  4. \(2x\)

Question 6: The marginal density of \(X\) is \(f(x)=6x(1 - x)\). Are \(X\) and \(Y\) independent?

Choose the correct answer:

  1. We do not have sufficient data to tell

  2. Yes

  3. No

  4. No, because \(X\) and \(Y\) are continuous variables.

Quiz 7: Discrete Random Variables Quiz 9: Correlation and covariance