Tagged: integral

Condition that a Function Be a Probability Density Function

Problem 756

Let $c$ be a positive real number. Suppose that $X$ is a continuous random variable whose probability density function is given by
f(x) = \begin{cases}
\frac{1}{x^3} & \text{ if } x \geq c\\
0 & \text{ if } x < c. \end{cases} \end{align*} (a) Determine the value of $c$.

(b) Find the probability $P(X> 2c)$.

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Matrix Representations for Linear Transformations of the Vector Space of Polynomials

Problem 71

Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less.
Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$.
For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation of $T$ with respect to the basis $B$. For $f(x)\in P_2(\R)$, define $T$ as follows.

(a) \[T(f(x))=\frac{\mathrm{d}^2}{\mathrm{d}x^2} f(x)-3\frac{\mathrm{d}}{\mathrm{d}x}f(x)\]

(b) \[T(f(x))=\int_{-1}^1\! (t-x)^2f(t) \,\mathrm{d}t\]

(c) \[T(f(x))=e^x \frac{\mathrm{d}}{\mathrm{d}x}(e^{-x}f(x))\]


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