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# Probability Density Functions Quiz

Created by
@ArdentParrot

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

The probability density function

### How is the expected value (or mean) of a continuous random variable $X$ calculated?

$E(X) = \int_{-\infty}^{\infty} x f(x) dx$

### Which expression represents the $k$-th moment of a continuous random variable $X$?

$E(X^k) = \int_{-\infty}^{\infty} x^k f(x) dx$

### 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

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