Probability Density Functions Quiz

Play an AI-generated podcast conversation about this lesson

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?

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

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

1 Signup and view all the answers

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

To calculate the area under the graph of $f(x)$ and determine the probability $P(a \leq X \leq b)$ 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 $f(x)$?

$P(a \leq X \leq b) = \int_a^b f(x) dx$ 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 $t \geq 0$?

$P(T \geq 2000) = 0.14$ 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?

$f(\theta) = \frac{1}{2\pi}$ for $\theta \in [0, 2\pi[$ 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})

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

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

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

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

$\int_{-\infty}^{\infty} xf_X(x) dx$ 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 $\alpha X + \beta Y$?

$\alpha E(X) + \beta E(Y)$ 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?

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

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

$\frac{4}{3}$ 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 $X$?

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

Probability Density Functions Quiz

Choose a study mode

Podcast

Questions and Answers