# 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.