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Excel in math and science
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Variance
Would you rather get paid $2 for flipping heads, or $6 for rolling a "1"? The expected value is the same ($1)...but the bets are different! Variance and standard deviation add color to probability.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
If the joint probability distribution function of \( X \) and \( Y \) is given as follows, what is \( 20 \text{Cov}(X,Y) ? \)
\[ \begin{matrix} p(X,Y) & X=0 & X=1 & X=2 \\ Y=0 & 0.1 & 0.1 & 0.2 \\ Y=1 & 0.2 & 0.2 & 0 \\ Y=2 & 0 & 0.1 & 0.1 \end{matrix} \]
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
For random variables \(X\) and \(Y,\) which of the following is the same as \[ \text{Cov}(X,Y)? \]
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
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If the joint probability density function of continuous random variables \( X \) and \( Y \) is \[ f(x,y) = \frac{1}{6} \left( 8 x + 4 y \right) \] with \( 0 \le x \le 1 \text{ and } 0 \le y \le 1, \) what is the covariance between \( X \) and \( Y? \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \(X\) and \(Y\) are independent random variables where \(E[X]=10 \) and \( E[Y]=8, \) what is \(E[XY]?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If the probability distributions of continuous random variables \( X \) and \( Y \) are independent of each other, which of the following is equal to \( \text{Cov}(X,Y)? \)