## MATH1015 Quizzes

Quiz 5: Integer-valued random variables
Question 1 Questions
In a production process, one fifth of the articles produced are defective. Ten articles from the production line are randomly selected and inspected. Let $X$ be the number of defective articles in this sample. If the distribution of $X$ is $B\left(n,p\right)$, what are the values of $n$ and $p$ ? Exactly one option must be correct)
 a) $n=10$, $p=0.1$ b) $n=10$, $p=0.2$ c) $n=10$, $p=1$ d) $n=0.2$, $p=10$

Choice (a) is incorrect
Choice (b) is correct!
With $p$ being the probability of an article being defective, $p=\frac{1}{5}=0.2$ and with $n$ being the sample size, $n=10$.
Choice (c) is incorrect
Choice (d) is incorrect
Suppose $X\sim B\left(10,0.2\right)$. Find $P\left(X\ge 3\right)$. Exactly one option must be correct)
 a) 0.8791 b) 0.1209 c) 0.3222 d) 0.6778 e) 0.6424

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$P\left(X\ge 3\right)=1-P\left(X\le 2\right)=1-0.6778=0.3222\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Choice (e) is incorrect
Suppose $X\sim B\left(10,0.2\right)$. Find the probability that $X$ is less than 2. Exactly one option must be correct)
 a) 0.6778 b) 0.3222 c) 0.6242 d) 0.3758

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$P\left(X<2\right)=P\left(X\le 1\right)=0.3758\phantom{\rule{0.3em}{0ex}}.$
Suppose that $X\sim B\left(10,0.2\right).$ What is the probability that $X=2$ ? Exactly one option must be correct)
 a) 0.6778 b) 0.302 c) 0.2684 d) 0.3758

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{rcll}P\left(X=2\right)& =& P\left(X\le 2\right)-P\left(X\le 1\right)& \text{}\\ & =& 0.6778-0.3758& \text{}\\ & =& 0.3020\phantom{\rule{0.3em}{0ex}}.& \text{}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Suppose that $X\sim B\left(10,0.2\right).$ What are the mean and variance of $X$ respectively ? Exactly one option must be correct)
 a) 1.6, 4 b) 2, 2.56 c) 2, 1.6 d) 2, 1.26 e) 1.6, 2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The mean of $X$ is $np=10×0.2=2$. The variance of $X$ is $np\left(1-p\right)=10×0.2×0.8=1.6$
Choice (d) is incorrect
Choice (e) is incorrect
Suppose $X\sim B\left(10,0.2\right)$. What is the probability that $X\le 2$ or $X\ge 8$ ? Exactly one option must be correct)
 a) 0.678 b) 0.6779 c) 0.678 d) 0.0002

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{rcll}P\left(X\le 2\phantom{\rule{1em}{0ex}}or\phantom{\rule{1em}{0ex}}X\ge 8\right)& =& P\left(X\le 2\right)+P\left(X\ge 8\right)& \text{}\\ & =& P\left(X\le 2\right)+\left(1-P\left(X\le 7\right)\right)& \text{}\\ & =& 0.6778+1-0.9999& \text{}\\ & =& 0.6779\phantom{\rule{0.3em}{0ex}}.& \text{}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Questions 7-9 use the following data.

In the following table, the probability of the random variable $X$ taking the values shown are: $\begin{array}{ccccccc}\hfill X\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ \hfill p\left(X\right)\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.3\hfill & \hfill 0.05\hfill & \hfill 0.05\hfill \end{array}$ What is the mean of the distribution ? Exactly one option must be correct)
 a) 1.95 b) 2.25 c) 2.15 d) 2.1

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The mean is $0×0.1+1×0.2+2×0.3+3×0.3+4×0.05+5×0.05=2.15$
Choice (d) is incorrect
Questions 7-9 use the following data.

In the following table, the probability of the random variable $X$ taking the values shown are

$\begin{array}{ccccccc}\hfill X\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ \hfill p\left(X\right)\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.3\hfill & \hfill 0.05\hfill & \hfill 0.05\hfill \end{array}$ What is the variance of the distribution ? Exactly one option must be correct)
 a) 6.15 b) 1.5275 c) 2.15 d) 4.6225 e) 4

Choice (a) is incorrect
Choice (b) is correct!
${\sum }_{i}ip\left(X=i\right)=2.15$, ${\sum }_{i}{i}^{2}p\left(X=i\right)=6.15$
$⇒Var\left(X\right)=6.15-{\left(2.15\right)}^{2}=1.5275\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Questions 7-9 use the following data.

In the following table, the probability of the random variable $X$ taking the values shown are

$\begin{array}{ccccccc}\hfill X\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲& ̲\\ \hfill p\left(X\right)\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.3\hfill & \hfill 0.05\hfill & \hfill 0.05\hfill \end{array}$ What is the standard deviation of the distribution ? Exactly one option must be correct)
 a) 1.46 b) 4 c) 2.3333 d) 1.2359

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The standard deviation $=\sqrt{Variance}=\sqrt{1.5275}=1.2359\phantom{\rule{0.3em}{0ex}}.$
An experiment consists of throwing a fair coin until the last two outcomes are both heads. Which of the following statements is true ? Exactly one option must be correct)
 a) The experiment is binomial. b) The probability that only two throws are required is $\frac{1}{4}$ . c) The probability that only four throws are required is $\frac{1}{16}$ . d) The probability that only eight throws are required is $\frac{1}{32}$ .

Choice (a) is incorrect
Choice (b) is correct!
The probability that only two throws are required is the probability of obtaining a head followed by a head, which is $\frac{1}{2}×\frac{1}{2}=\frac{1}{4}$.
Choice (c) is incorrect
Choice (d) is incorrect