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Exercises

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 $E(X)$?