PROBABILITIES Course

Exercises

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This is a batch of exercises on continuous probabilities (density function, expected value, variance, moment generating function, characteristic function).


Exercise 1 (density function)

Verify that $f(x) = x * e^{\frac{-x^2}{2}}$ with $x \gt 0$ is a density function.

1. $\forall{x},\ f(x) \ge 0$

$x \gt 0$ and $e^t \ge 0$ so $f(x) \ge 0$.

2. $f_X$ is continuous

Both $x$ and $e^t$ are continuous.

3. $\int_{-\infty}^{+\infty} f_X(x)dx = 1$ that we can simplify since $x \gt 0$ giving us $\int_{0}^{+\infty} f(x) dx = 1$.

\[ \begin{split} \int_{0}^{+\infty} x * e^{\frac{-x^2}{2}} dx = 1 \\ \Leftrightarrow [-e^{\frac{-x^2}{2}}]_{0}^{+\infty} = 1 \\ \Leftrightarrow -0 - -1 = 1 \end{split} \]

So $f(x)$ is a density function.


Exercise 2 (density function)

Verify that $f(x) = 1 - |x|$ with $x \in [[-1,1]]$ is a density function.

We can consider this a sum of two integrals from 0 to 1 because of the $|x|$ (absolute value) giving us $x \in [[0,1]]$.

@ 2 * \int_{0}^{1} 1 - x @

1. $\forall{x},\ f(x) \ge 0$

Since $x \in [0,1]$ then $1-x \ge 0$.

2. $f_X$ is continuous

Both $x$ and $1-x$ are continuous.

3. $\int_{-\infty}^{+\infty} f_X(x)dx = 1$

\[ \begin{split} 2 * \int_{0}^{1} 1 - x = 1 \\ \Leftrightarrow 2 * [x - \frac{x^2}{2} ]_{0}^{1} = 1 \\ \Leftrightarrow 2 * (1 - 0.5 - (0 - 0)) = 1 \\ \Leftrightarrow 2 * 0.5 = 1 \end{split} \]

So $f(x)$ is a density function.