7.1 Exercises

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 for $\lambda = 0.5$ and $\lambda = 1$..
3. Compute $\Pr(X > 3)$ when $\lambda = 1$.
4. Compute the mean and variance of $X$.

2. 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. Show that $f_W(w)$ is a pmf.
3. Determine and sketch the CDF.
4. Compute $\Pr(W>2)$.
5. Compute the mean and variance of $W$.

3. 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. Find $\Pr(X \le 2)$.
   4. Find $\Pr(2 < X < 4)$.

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.

5. 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 $k$.

6. 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 $i$ with variance $\sigma^2$.

7. Show that the mean and the variance of the Poisson distribution are both $\mu$.

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

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Copyright © Peter K. Dunn, 2021