Statistics

Randomly chose a coordinate (x, y, z) on the unit sphere. What is the variance of the z-coordinate?
Suppose you roll three fair 100-sided die. What is the expected value of the lowest roll?
A donut shop has 25 different bakers, each of whom bake donuts. Each baker's output is independent from the others. It so happens that the standard deviation in donut output for each baker is 10. What is the standard deviation in the average number of donuts made by a baker?
Imagine you are given a standard deck of 52 cards (half of the cards are red and the other half are black). A run is defined as a block of cards that are drawn consecutively and all have the same color. As an example, BBRRBBB has 3 runs.
1. Find the expected number of runs in a shuffled deck of cards.
Suppose a game has you roll a fair six-sided die repeatedly, until you first encounter a number which has previously appeared. Let r denote number of times the die was rolled, and $p_{r}$ the probability of rolling r times.
1. Calculate $p_{3}$
2. What is the value of $p_{1} + p_{2} + \dots + p_{10}$
3. What is $p_{5}$ given that this iteration of the game has taken at least 3 rolls?
What is the variance of a uniform random variable X with probability density function $f (x) = 1$ for $0 \leq x \leq 1$ and zero elsewhere?
A certain disease has an incidence rate of 2%. If the false negative rate is 10% and the false positive rate is 1%
1. What is the probability that a person who tests positive actually has the disease?
2. A medical research lab has concocted a new drug that they claim will cure 80% of people suffering from a novel disease. After examining the drug, the FDA believes that their claims regarding the efficacy of the drug are inflated. To validate their hypothesis, the FDA administers the drug to 20 people with the novel disease and observes, XX, the number of individuals cured by the drug. More formally, the FDA is conducting the following test:
  1. $H_{0} : p = 0.8$ and $H_{a} : p < 0.8$ Assuming a rejection region of $x \leq 12$ is used, what is the significance level, $α$ , for this study?
  2. Assuming a rejection region of $X \leq 12$ , what is the power of the test when $p = 0.6$ ?

Answer

1. Variance of $z$ -coordinate on the unit sphere

By symmetry of the uniform distribution on the unit sphere, $E [X^{2}] = E [Y^{2}] = E [Z^{2}]$ . Since $X^{2} + Y^{2} + Z^{2} = 1$ :

$E [Z^{2}] = \frac{1}{3}$

Also by symmetry, $E [Z] = 0$ , so:

$Var (Z) = E [Z^{2}] - (E [Z])^{2} = \frac{1}{3}$

2. Expected value of the minimum of three $d 100$ rolls

Using the tail-sum formula for non-negative integer random variables:

$E [min] = k = 1 \sum 100 P (min \geq k) = k = 1 \sum 100 (\frac{100 - k + 1}{100})^{3} = \frac{1}{10 0 ^{3}} j = 1 \sum 100 j^{3}$

Using $\sum_{j = 1}^{n} j^{3} = (\frac{n ( n + 1 )}{2})^{2}$ :

$E [min] = \frac{1}{1 0 ^{6}} (\frac{100 \cdot 101}{2})^{2} = \frac{505 0 ^{2}}{1 0 ^{6}} = \frac{25 , 502 , 500}{1 , 000 , 000} = 25.5025$

3. Standard deviation of the average baker output

The standard deviation of a sample mean of $n$ independent observations is $σ / n$ :

$σ_{\overset{ˉ}{X}} = \frac{σ}{n} = \frac{10}{25} = 2$

4. Expected number of runs in a shuffled deck

Define indicators $X_{i}$ where $X_{i} = 1$ if position $i$ starts a new run. Always $X_{1} = 1$ . For $i \geq 2$ , $X_{i} = 1$ if positions $i - 1$ and $i$ differ in color.

$P (X_{i} = 1) = \frac{2 \cdot 26 \cdot 26}{52 \cdot 51} = \frac{26}{51}$

By linearity of expectation:

$E [runs] = 1 + 51 \cdot \frac{26}{51} = 1 + 26 = 27$

5. Repeated rolling

(a) $p_{3}$ requires the first two rolls to differ and the third to match one of them:

$p_{3} = 1 \cdot \frac{5}{6} \cdot \frac{2}{6} = \frac{10}{36} = \frac{5}{18}$

(b) By the pigeonhole principle, with only 6 distinct values, the game must end by the 7th roll. Also $p_{1} = 0$ . So $p_{2} + p_{3} + \dots + p_{7} = 1$ and all other $p_{r} = 0$ :

$p_{1} + p_{2} + \dots + p_{10} = 1$

(c) First compute $p_{5}$ (rolls 1–4 all distinct, roll 5 matches one of them):

$p_{5} = 1 \cdot \frac{5}{6} \cdot \frac{4}{6} \cdot \frac{3}{6} \cdot \frac{4}{6} = \frac{240}{1296} = \frac{5}{27}$

The event "at least 3 rolls" is the complement of $r = 2$ , so $P (r \geq 3) = 1 - \frac{1}{6} = \frac{5}{6}$ .

$P (r = 5 ∣ r \geq 3) = \frac{p _{5}}{P ( r \geq 3 )} = \frac{5/27}{5/6} = \frac{2}{9}$

6. Variance of $X \sim Uniform (0, 1)$

$E [X] = \int_{0}^{1} x d x = \frac{1}{2}, E [X^{2}] = \int_{0}^{1} x^{2} d x = \frac{1}{3}$

$Var (X) = \frac{1}{3} - (\frac{1}{2})^{2} = \frac{1}{12}$

7. Disease testing

(a) Let $D$ = has disease, $+$ = positive test. Given $P (D) = 0.02$ , $P (+ ∣ D) = 0.9$ , $P (+ ∣ D^{c}) = 0.01$ :

$P (D ∣ +) = \frac{P ( + ∣ D ) P ( D )}{P ( + ∣ D ) P ( D ) + P ( + ∣ D ^{c} ) P ( D ^{c} )} = \frac{0.9 \cdot 0.02}{0.9 \cdot 0.02 + 0.01 \cdot 0.98}$

$= \frac{0.018}{0.0278} = \frac{90}{139} \approx 0.6475$

(b)(i) Under $H_{0}$ , $X \sim Binomial (20, 0.8)$ . Equivalently let $Y = 20 - X \sim Binomial (20, 0.2)$ , so ${X \leq 12} = {Y \geq 8}$ :

$α = P (X \leq 12 ∣ p = 0.8) = k = 0 \sum 12 (k 20) (0.8)^{k} (0.2)^{20 - k} \approx 0.0321$

(b)(ii) Under $p = 0.6$ , $X \sim Binomial (20, 0.6)$ :

$Power = P (X \leq 12 ∣ p = 0.6) = k = 0 \sum 12 (k 20) (0.6)^{k} (0.4)^{20 - k} \approx 0.5841$

Reference (and explanation)

StatQ