**1.** Let the distribution of $X$ be

$~~~~~~~~~~~ x$ | 1 | 2 | 3 |
---|---|---|---|

$P(X=x)$ | $0.2$ | $0.5$ | $0.3$ |

Find $E(X)$ and $Var(X)$.

**2.** A person is picked at random from a population. Let $Y$ be the year in which the person was born, and suppose $E(Y) = 1997$ and $SD(Y) = 2$. Define the person's age in 2019 to be $X = 2019 - Y$. Find $E(X)$ and $SD(X)$.

**3.**
In each part, construct the distribution of a random variable $X$ that satisfies the conditions given. A great way to construct examples is to keep them as simple as possible: random variables with just a couple of possible values will do the job.

**a)** $E(X) = 10$, $SD(X) = 0$

**b)** $E(X) = 10$, $SD(X) = 20$.

**c)** $E(X) = 10$, $SD(X) = 20$, $P(X \ge 0) = 1$.

**4.**
Let $X$ have distribution

$~~~~~~~~~~~ x$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|

$P(X=x)$ | $0.4$ | $0.1$ | $0.1$ | 0.4 |

Let $Y$ have distribution

$~~~~~~~~~~~ y$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|

$P(Y=Y)$ | $0.1$ | $0.4$ | $0.4$ | $0.1$ |

In each part, say which of the two quantities is bigger (if any) and explain why.

**a)** $E(X)$, $E(Y)$

**b)** $SD(X)$, $SD(Y)$

**5.**
Let $p \in (0, 1)$ and let $X$ be the number of spots showing on a flattened die that shows its six faces according to the following chances:

- $P(X=1) ~ = ~ P(X=6)$
- $P(X=2) ~ = ~ P(X=3) ~ = ~ P(X=4) ~ = ~ P(X=5)$
- $P(X = 1 ~ \text{ or } ~ 6) = p$

Find $SD(X)$ and explain why it is an increasing function of $p$.

**6.**
Ages in a population have a mean of 40 years. Let $X$ be the age of a person picked at random from the population.

**a)** If possible, find $P(X \ge 80)$. If it's not possible, explain why, and find the best upper bound you can based on the information given.

**b)** Suppose you are told in addition that the SD of the ages is 15 years. What can you say about $P(10 < X < 70)$?

**c)** With the information as in Part **b**, what can you say about $P(10 \le X \le 70)$?

**7.**
Scores on a test have an average of 60 and a standard deviation of 12. Let $S$ be a score picked at random. Find the best lower and upper bounds you can on $P(S \ge 90)$.

**8.**
Let $X$ be a non-negative random variable, and let a "median" $m$ be such that $P(X \ge m) = 0.5$. Find a bound for the mean $E(X)$ in terms of the median $m$.

**9.**
Let $X$ be a random variable with $E(X) = 20$ and $SD(X) = 4$. Use Markov's inequality to find an upper bound for $P(X^2 \ge 1000)$.

**10.**
Let $X$ be a random variable.

**a)** If you know $E(X)$ and $E(X^2)$, can you find $SD(X)$?

**b)** If you know $E(X)$ and $SD(X)$, can you fnd $E(X^2)$?

**c)** If you know $SD(X)$ and $E(X^2)$, can you find $E(X)$?