16 Questions
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?
The difference between the cumulative distribution functions evaluated at $b$ and $a$
What is the value of $\int_{-\infty}^{\infty} f(x) dx$ for a valid probability density function $f(x)$?
1
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 $P(a \leq X \leq b)$
What is the formula for calculating the probability $P(a \leq X \leq b)$ for a continuous random variable $X$ with probability density function $f(x)$?
$P(a \leq X \leq b) = \int_a^b f(x) dx$
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 $t \geq 0$?
$P(T \geq 2000) = 0.14$
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 $\theta \in [0, 2\pi[$
What is the relationship between the probability density function $f(x)$ and the total probability $P(X \in \mathbb{R})
$\int_{-\infty}^{\infty} f(x) dx = 1$
What is the probability density function of the standard Cauchy distribution?
$\frac{1}{\pi(1 + x^2)}$
What is the expected value of a continuous random variable $X$ with probability density function $f_X(x)$?
$\int_{-\infty}^{\infty} xf_X(x) dx$
If $X$ and $Y$ are two continuous random variables with expected values $E(X)$ and $E(Y)$ respectively, what is the expected value of $\alpha X + \beta Y$?
$\alpha E(X) + \beta E(Y)$
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
What is the expected value of the square of the randomly chosen number in the interval $[0, 2]$?
$\frac{4}{3}$
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 $X$?
$E(g(X)) = \int_{-\infty}^{\infty} g(x)f_X(x) dx$
Test your knowledge on probability density functions by answering questions related to understanding the area under the graph of a function, calculating probabilities using density functions, and interpreting propositions related to continuous random variables.
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