# 7 Test yourself

- Consider a rv \(X\) with probability function: \[ f_X(x) = \lambda \exp(-\lambda x)\qquad\text{where $\lambda>0$} \] for \(x>0\).

- Decide if \(X\) is discrete or continuous.
- Determine and sketch the CDF.
- Compute \(\Pr(x>3)\).
- Compute the mean and variance of \(X\).

- Consider a rv \(W\) with probability function \[ f_W(w) = 0.6(0.4)^{w-1} \] when \(w = 1,2,\dots\) and is zero elsewhere.

- Decide if \(W\) is discrete or continuous.
- Determine and sketch the CDF.
- Compute \(\Pr(W>2)\).
- Compute the mean and variance of \(W\).

Suppose \(X\) is a continuous rv and \(f_X(x) = k\exp(-x/4)\) for \(x>0\).

- Find \(k\) so that \(f_X(x)\) is a PDF.
- Find and sketch \(F_X(x)\).
- \(\Pr(X \le 2)\).
- Find \(\Pr(2<X<4)\).

The joint PDF of \(X\) and \(Y\) is \[ f_{X,Y}(x,y) = \left\{ \begin{array}{ll} k(3x^2 + xy) & \text{for $0\le x \le 1$ and $0 \le y \le 2$}\notag\\ 0 & \text{elsewhere} \end{array} \right. \] Find the marginal PDFs.

Consider the two continuous random variables \(Y\) and \(Z\) with joint probability function \[ f_{Y,Z}(y,z) = k(y+z) \] for \(0<y<z<1\).

- Sketch the sample space.
- Find a value for \(k\).

Show that \(\text{var}[\bar{X}] = \sigma^2/n\), where \(\bar{X} = (x_1 + X_2 + \cdots + X_n)/n\) and \(X_i\) are independent and have identical distributions for all~\(i\).

Show that the mean and the variance of the Poisson distribution are both \(\mu\).

Find the mean and variance of an exponential distribution.