Podcast
Questions and Answers
What does the symbol $f(x)$ represent in the continuous case?
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?
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 $X$?
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?
What does the expression $\int_{a}^{b} f(x) dx$ represent?
What is the value of $\int_{-\infty}^{\infty} f(x) dx$ for a valid probability density function $f(x)$?
What is the value of $\int_{-\infty}^{\infty} f(x) dx$ for a valid probability density function $f(x)$?
What is the primary purpose of the probability density function $f(x)$ described in the text?
What is the primary purpose of the probability density function $f(x)$ described in the text?
What is the formula for calculating the probability $P(a \leq X \leq b)$ for a continuous random variable $X$ with probability density function $f(x)$?
What is the formula for calculating the probability $P(a \leq X \leq b)$ for a continuous random variable $X$ with probability density function $f(x)$?
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 $t \geq 0$?
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 $t \geq 0$?
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?
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?
What is the relationship between the probability density function $f(x)$ and the total probability $P(X \in \mathbb{R})
What is the relationship between the probability density function $f(x)$ and the total probability $P(X \in \mathbb{R})
What is the probability density function of the standard Cauchy distribution?
What is the probability density function of the standard Cauchy distribution?
What is the expected value of a continuous random variable $X$ with probability density function $f_X(x)$?
What is the expected value of a continuous random variable $X$ with probability density function $f_X(x)$?
If $X$ and $Y$ are two continuous random variables with expected values $E(X)$ and $E(Y)$ respectively, what is the expected value of $\alpha X + \beta Y$?
If $X$ and $Y$ are two continuous random variables with expected values $E(X)$ and $E(Y)$ respectively, what is the expected value of $\alpha X + \beta Y$?
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?
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?
What is the expected value of the square of the randomly chosen number in the interval $[0, 2]$?
What is the expected value of the square of the randomly chosen number in the interval $[0, 2]$?
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 $X$?
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 $X$?
Flashcards
Probability Density Function (continuous case)
Probability Density Function (continuous case)
A function that describes the relative likelihood of a continuous random variable taking on a given value.
Expected Value (continuous)
Expected Value (continuous)
The average value of a continuous random variable, calculated by integrating the product of the variable and its probability density function.
k-th moment (continuous)
k-th moment (continuous)
The expected value of the k-th power of a continuous random variable.
∫f(x)dx (a to b)
∫f(x)dx (a to b)
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∫f(x)dx (inf to inf)
∫f(x)dx (inf to inf)
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Purpose of PDF
Purpose of PDF
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P(a ≤ X ≤ b)
P(a ≤ X ≤ b)
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Standard Cauchy PDF
Standard Cauchy PDF
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Expected Value Formula (continuous)
Expected Value Formula (continuous)
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E(αX + βY)
E(αX + βY)
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Uniform PDF [0,2]
Uniform PDF [0,2]
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E(X^2) [0,2]
E(X^2) [0,2]
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E[g(X)]
E[g(X)]
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PDF relationship to total probability
PDF relationship to total probability
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