7 Test yourself

  1. Consider a rv \(X\) with probability function: \[ f_X(x) = \lambda \exp(-\lambda x)\qquad\text{where $\lambda>0$} \] for \(x>0\).
  1. Decide if \(X\) is discrete or continuous.
  2. Determine and sketch the CDF.
  3. Compute \(\Pr(x>3)\).
  4. Compute the mean and variance of \(X\).
  1. 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.
  1. Decide if \(W\) is discrete or continuous.
  2. Determine and sketch the CDF.
  3. Compute \(\Pr(W>2)\).
  4. Compute the mean and variance of \(W\).
  1. Suppose \(X\) is a continuous rv and \(f_X(x) = k\exp(-x/4)\) for \(x>0\).

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

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

  1. Sketch the sample space.
  2. Find a value for \(k\).
  1. 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~\(i\).

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

  3. Find the mean and variance of an exponential distribution.