# Exercises

# 6.5. 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 \(E(X)\)?