## Questions and Answers

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?

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

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

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

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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 $t \geq 0$?

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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?

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

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

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

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

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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?

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

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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 $X$?

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