Problem 2 Page 321

Let X be a continuous random variable with values uniformly distributed over the interval [0, 20]. (a) Find the mean and variance of X. (b) \(P(|X - 10| \ge 2)\), \(P(|X - 10| \ge 5)\), \(P(|X - 10| \ge 9)\) and \(P(|X - 10| \ge 20)\) exactly. How do your answers compare with those of Exercise 1? How good is Chebyshev’s Inequality in this case?

Answer :

1. Chebyshev’s Inequality mean and variance are

\(E(X)={(a+b)}/2\)

\(Var(X)={(b - a)^2}/12\)

So Mean \(E(X)\) is :

print((0+20)/2)

## [1] 10

and Variance \(Var(X)\) is :

print(((20-0)**2)/12)

## [1] 33.33333

2. \(P(|X - 10| \ge 2)\) =\(1-P(|X - 10| \lt 2)\)

=\(1-P(-2 \lt (X -10) \lt 2)\)

=\(1-P(-2+10 \lt X \lt 2 +10)\)

=\(1-P(8 \lt X \lt 12)\)

=\(1-[P(X \lt 12) - P(X \le 8)]\)

=\(1-[12/20 - 8/20]\)

print(1-((12/20) - (8/20)))

## [1] 0.8

\(P(|X - 10| \ge 5)\)

=\(1-P(|X - 10| \lt 5)\)

=\(1-P(-5 \lt (X-10) \lt 5)\)

=\(1-P(-5+10 \lt X \lt 5+10)\)

=\(1-P(5 \lt X \lt 15)\)

=\(1-[P(X \lt 15) - P(X \le 5)]\)

print(1-((15/20) - (5/20)))

## [1] 0.5

\(P(|X - 10| \ge 9)\)

=\(1-P(|X - 10| \lt 9)\)

=\(1-P(-9 \lt (X-10) \lt 9)\)

=\(1-P(-9+10 \lt X \lt 9+10)\)

=\(1-P(1 \lt X \le 19)\)

=\(1-[P(X \lt 19) - P(X \le 1)]\)

print(1-((19/20) - (1/20)))

## [1] 0.1

\(P(|X - 10| \ge 20)\)

=\(1-P(|X - 10| \lt 20)\)

=\(1-P(-20 \lt (X-10) \lt 20)\)

=\(1-P(-20+10 \lt X \lt 20+10)\)

=\(1-P(-10 \lt X \le 30)\)

=\(1-P(0 \lt X \le 20)\)

=\(1-1\)

print(1-1)

## [1] 0

How do your answers compare with those of Exercise 1? How good is Chebyshev’s Inequality in this case?

Answer :

From problem 1 :

1. \(\epsilon=2\), the upper bound is \(1\) from this problem upper bound is \(0.8\)
2. \(\epsilon=5\), the upper bound is \(1\) from this problem upper bound is \(0.5\)
3. \(\epsilon=9\), the upper bound is \(\approx 0.4115\) from this problem upper bound is \(0.1\)
4. \(\epsilon=20\), the upper bound is \(\approx 0.0833\) from this problem upper bound is \(0.0\)

Chebyshev estimates are in general not very accurate. We see variance for \(\epsilon=2, 5, 9, 20\)