Probability Density Functions Quiz
16 Questions
1 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What does the symbol $f(x)$ represent in the continuous case?

  • The moment generating function
  • The probability density function (correct)
  • The cumulative distribution function
  • The probability mass function
  • How is the expected value (or mean) of a continuous random variable $X$ calculated?

  • $E(X) = \int_{-1}^{1} x f(x) dx$
  • $E(X) = \sum_{i} x_i P(X = x_i)$
  • $E(X) = \int_{-\infty}^{\infty} x f(x) dx$ (correct)
  • $E(X) = \int_{0}^{1} x f(x) dx$
  • Which expression represents the $k$-th moment of a continuous random variable $X$?

  • $E(X^k) = \int_{0}^{1} x^k f(x) dx$
  • $E(X^k) = \int_{-1}^{1} x^k f(x) dx$
  • $E(X^k) = \sum_{i} x_i^k P(X = x_i)$
  • $E(X^k) = \int_{-\infty}^{\infty} x^k f(x) dx$ (correct)
  • What does the expression $\int_{a}^{b} f(x) dx$ represent?

    <p>The difference between the cumulative distribution functions evaluated at $b$ and $a$</p> Signup and view all the answers

    What is the value of $\int_{-\infty}^{\infty} f(x) dx$ for a valid probability density function $f(x)$?

    <p>1</p> Signup and view all the answers

    What is the primary purpose of the probability density function $f(x)$ described in the text?

    <p>To calculate the area under the graph of $f(x)$ and determine the probability $P(a \leq X \leq b)$</p> Signup and view all the answers

    What is the formula for calculating the probability $P(a \leq X \leq b)$ for a continuous random variable $X$ with probability density function $f(x)$?

    <p>$P(a \leq X \leq b) = \int_a^b f(x) dx$</p> Signup and view all the answers

    What is the probability that a computer will still be functioning after 2000 hours of usage, given the probability density function $f(t) = \frac{1}{1000}e^{-t/1000}$ for $t \geq 0$?

    <p>$P(T \geq 2000) = 0.14$</p> Signup and view all the answers

    What is the probability density function $f(\theta)$ that describes the direction of emission of an alpha particle in two dimensions, according to the text?

    <p>$f(\theta) = \frac{1}{2\pi}$ for $\theta \in [0, 2\pi[$</p> Signup and view all the answers

    What is the relationship between the probability density function $f(x)$ and the total probability $P(X \in \mathbb{R})

    <p>$\int_{-\infty}^{\infty} f(x) dx = 1$</p> Signup and view all the answers

    What is the probability density function of the standard Cauchy distribution?

    <p>$\frac{1}{\pi(1 + x^2)}$</p> Signup and view all the answers

    What is the expected value of a continuous random variable $X$ with probability density function $f_X(x)$?

    <p>$\int_{-\infty}^{\infty} xf_X(x) dx$</p> Signup and view all the answers

    If $X$ and $Y$ are two continuous random variables with expected values $E(X)$ and $E(Y)$ respectively, what is the expected value of $\alpha X + \beta Y$?

    <p>$\alpha E(X) + \beta E(Y)$</p> Signup and view all the answers

    Suppose we randomly choose a number in the interval $[0, 2]$ and compute its square. What is the probability density function of the chosen number?

    <p>$\frac{1}{2}$ if $x \in [0, 2]$, 0 otherwise</p> Signup and view all the answers

    What is the expected value of the square of the randomly chosen number in the interval $[0, 2]$?

    <p>$\frac{4}{3}$</p> Signup and view all the answers

    What is the relationship between the probability density function $f_X(x)$ and the expected value $E(g(X))$ of a function $g$ of a continuous random variable $X$?

    <p>$E(g(X)) = \int_{-\infty}^{\infty} g(x)f_X(x) dx$</p> Signup and view all the answers

    Use Quizgecko on...
    Browser
    Browser