Exercises
Statistics I
Week 12 Graded Assignment

Week 12 Graded Assignment


Statistics Graded Assignment

Here are all the questions and their solutions from the Week_12_Graded_Solution.pdf (Statistics for Data Science-1, Week 12 Graded Assignment, Continuous random variables)12:

1. Uniformly Distributed Random Variable

Question: Let a random variable be uniformly distributed over $[a, b]$ with expectation $e$ and variance $v$ respectively. Find the value of $ab$.

Answer: $ab = e^2 - 3v$

Solution: Given:

$$ E[X] = \frac{b+a}{2} = e \implies (b+a)^2 = 4e^2 $$$$ \text{Var}(X) = \frac{(b-a)^2}{12} = v \implies (b-a)^2 = 12v $$$$ (b+a)^2 - (b-a)^2 = 4ab = 4e^2 - 12v \implies ab = e^2 - 3v $$

Example: If $e=6$, $v=4$:

$$ ab = 36 - 12 = 24 $$

2. Probability Density Function (Quadratic)

Question: The probability density function of a random variable $X$ is given by

$$ f(x) = \begin{cases} kx^2 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases} $$

Find $P(m < X < n)$. Options: a. $n^2 - m^2$ b. $\frac{n^3 - m^3}{9}$ c. $\frac{n^2 - m^2}{9}$ d. $n^3 - m^3$

Answer: d. $n^3 - m^3$

Solution:

$$ \int_0^1 kx^2 dx = 1 \implies k = 3 $$$$ P(m < X < n) = \int_m^n 3x^2 dx = n^3 - m^3 $$

Example: If $m=0.1$, $n=0.7$:

$$ P = 0.7^3 - 0.1^3 = 0.342 $$

3. Uniform Distribution (Milk Production)

Question: The amount of milk produced every day by a dairy is uniformly distributed between 100 litres and 120 litres. What is the probability that the amount of milk produced is more than $m$ litres given that on that day, more than $n$ litres of milk was produced? (Answer correct to 2 decimal places)

Answer:

$$ \frac{120 - m}{120 - n} $$

Solution:

$$ P(X > m \mid X > n) = \frac{1 - P(X \leq m)}{1 - P(X \leq n)} = \frac{120 - m}{120 - n} $$

Example: If $m=115$, $n=110$:

$$ P = \frac{5}{10} = 0.5 $$

4. Uniform Distribution (Train Arrival)

Question: The time that Jan Shatabdi express will reach the Delhi station is uniformly distributed between $x$ p.m. and $y$ p.m. What is the probability that the train reaches Delhi exactly at $z$ p.m.?

Answer: 0

Solution: For a continuous uniform distribution, the probability at any exact point is zero.

5. Exponential Distribution (Restaurant Service)

Question: The length of time for one person to be served at a restaurant is exponentially distributed, with an expected time of $t$ minutes. If 5 persons arrived at the restaurant, what is the probability that at least 2 of them will be served in more than $p$ minutes? (Assume independence)

Options: a. $1 - (1 - e^{-p/t})^5 - 5e^{-p/t}(1 - e^{-p/t})^4$ b. $1 - (e^{-p/t})^5 - [5(e^{-p/t})^4(1 - e^{-p/t})]$ c. $1 - (1 - e^{-pt})^5 - [5e^{-pt}(1 - e^{-pt})^4]$ d. $1 - (e^{-pt})^5 - [5e^{-4pt}(1 - e^{-pt})]$

Answer: a. $1 - (1 - e^{-p/t})^5 - 5e^{-p/t}(1 - e^{-p/t})^4$

Solution:

$$ P(X > p) = e^{-p/t} $$

Probability at least 2 out of 5 take more than $p$ minutes:

$$ P(\text{at least 2}) = 1 - P(0) - P(1) $$$$ P(0) = (1 - e^{-p/t})^5, \quad P(1) = 5e^{-p/t}(1 - e^{-p/t})^4 $$$$ P(\text{at least 2}) = 1 - (1 - e^{-p/t})^5 - 5e^{-p/t}(1 - e^{-p/t})^4 $$

6. Piecewise PDF

Question: Probability density function of a random variable $X$ is given by

$$ f(x) = \begin{cases} 0.2x & \text{if } 0 \leq x \leq 1 \\ 0.2 & \text{if } 1 \leq x \leq 2 \\ 0.2x - 0.2 & \text{if } 2 \leq x \leq 3 \\ 0.4 & \text{if } 3 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases} $$

Find the probability that $0 < X < 2.5$. (Answer up to three decimal points)

Answer: 0.425 (accepted range: [0.421, 0.429])

Solution:

$$ P(0 < X < 2.5) = \int_0^1 0.2x\,dx + \int_1^2 0.2\,dx + \int_2^{2.5} (0.2x - 0.2)\,dx = 0.425 $$

7. Exponential Distribution (Visitor Arrival)

Question: In a certain exhibition, the time for the next visitor to come is exponentially distributed with a standard deviation of $t$ minutes. What will be the expected time (in minutes) for two visitors to arrive if one comes after the other? (Assume independence)

Options: a. $\frac{1}{t}$ b. $2t$ c. $t$ d. $\frac{2}{t}$

Answer: b. $2t$

Solution:

$$ \text{Mean} = t, \quad \text{For two independent events: } 2t $$

8. Uniform Distribution (Badminton Match)

Question: The total duration (in minutes) of a badminton match in the Premier Badminton League (PBL) is uniformly distributed between $[a, b]$ with variance 12 square minutes. The probability that a match will last at most $m$ minutes is $\frac{1}{p}$. Find the expected time duration (in minutes) of a badminton match. (Answer correct to 1 decimal place)

Solution:

$$ \frac{(b-a)^2}{12} = 12 \implies b-a = 12 $$$$ P(X \leq m) = \frac{m-a}{b-a} = \frac{1}{p} \implies m-a = \frac{12}{p} $$$$ a = m - \frac{12}{p}, \quad b = m - \frac{12}{p} + 12 $$$$ E[X] = \frac{a+b}{2} = m - \frac{12}{p} + 6 $$

Example: If $m=42$, $p=6$:

$$ a = 40, \quad b = 52, \quad E[X] = 46 $$

9. Exponential Distribution (Customer Care)

Question: The probability density function of the time $X$ (in minutes) between calls at the customer care is given by

$$ f(x) = \begin{cases} \frac{1}{m} e^{-x/m} & \text{if } 0 \leq x < \infty \\ 0 & \text{otherwise} \end{cases} $$

Find the probability that time between calls exceeds the mean time.

Options: a. $1 - \frac{1}{e}$ b. $1 - e$ c. $\frac{1}{e}$ d. $e$

Answer: c. $\frac{1}{e}$

Solution:

$$ P(X \geq m) = e^{-1} = \frac{1}{e} $$

10. Exponential Distribution (Light Bulb Lifetime)

Question: The lifetime of a light bulb is exponentially distributed with a mean life of $n$ months. If there are $p%$ chances that a light bulb will last at most $t$ months, what is the value of $t$?

Options: a. $n \ln\left(\frac{100-p}{100}\right)$ b. $n \ln\left(\frac{100}{100-p}\right)$ c. $\frac{1}{n} \ln\left(\frac{100-p}{100}\right)$ d. $\frac{1}{n} \ln\left(\frac{100}{100-p}\right)$

Answer: b. $n \ln\left(\frac{100}{100-p}\right)$

Solution:

$$ P(X \leq t) = \frac{p}{100} \implies 1 - e^{-t/n} = \frac{p}{100} $$$$ e^{-t/n} = \frac{100-p}{100} \implies t = n \ln\left(\frac{100}{100-p}\right) $$

Example: If $n=18$, $p=60$:

$$ t = 18 \ln(2.5) $$

This covers all questions and solutions from the PDF12.

⁂

  1. Week_12_Graded_Solution.pdf ↩︎ ↩︎

  2. https://www.studocu.com/in/document/indian-institute-of-technology-madras/iitm-online-degree-data-science-and-programming/week-12-graded-solution/102388299 ↩︎ ↩︎

Week 11 Graded AssignmentWeek 4 Statistics Graded Assignment

More Questions

Profile SVG
Add Chart.js in HUGO 📈 Statistics Graded Assignment

This is an inline equation: $E = mc^2$ and a block equation: $$a^2 + b^2 = c^2$$ Find $x^2$. % KaTeX inline notation Inline notation: % KaTeX inline notation Inline notation: \(\varphi = \dfrac{1+\sqrt5}{2}= 1.6180339887…\)

Profile SVG
Week 1 Graded Assignment Statistics Graded Assignment

Questions and Solutions 🧠 1. Identify the sample and population. Question: The education minister wants to know the status of campus placements of B.Tech students in different engineering institutes of India. An analyst did a survey on the randomly selected four IITs of India and analysed the status of campus placements. Based on the information given, answer the question. Options: (a) The sample consists of all the engineering institutes of India and the population consists of randomly selected four IITs of India. (b) The sample consists of all the IITs of India and the population consists of all the engineering institutes of India. (c) The sample consists of all IITs of India and the population consists of randomly selected four IITs of India. (d) The sample consists of four randomly selected IITs of India and the population consists of all the engineering institutes of India.

Profile SVG
Week 2 Graded Assignment Statistics Graded Assignment

1. Which of the following statements is/are incorrect? Options: (a) To represent the share of a particular category, bar chart is the most appropriate graphical representation. (b) The multiplication of the total number of observations and relative frequency of a particular observation should be equal to the frequency of that observation. (c) Mean can be defined for a categorical variable. (d) Mode of a categorical variable is the widest slice in a pie chart.

Profile SVG
Week 3 Graded Assignment Statistics Graded Assignment

1. The numbers a, b, c, d have frequencies (x + 6), (x + 2), (x − 3) and x respectively. If their mean is m, find the value of x. (Enter the value as next highest integer) Solution: $$ \frac{a(x + 6) + b(x + 2) + c(x − 3) + dx}{(x + 6) + (x + 2) + (x − 3) + x} = m $$$$ \frac{ax + 6a + bx + 2b + cx − 3c + dx}{4x + 5} = m $$$$ ax + bx + cx + dx + 6a + 2b − 3c = m(4x + 5) = (4m)x + 5m $$$$ (a + b + c + d − 4m)x = 5m − 6a − 2b + 3c $$$$ x = \frac{5m − 6a − 2b + 3c}{a + b + c + d − 4m} $$Suppose, we substitute values of a, b, c, d and m as 2, 7, 9, 17 and 6.88 respectively,

Profile SVG
Week 4 Graded Assignment Statistics Graded Assignment

Questions 1–6: Sales Data Analysis Context: The phone brands OnePlus, Vivo, and Oppo are owned by BBK Electronics. Table 4.1.G represents sales (in Lakhs) of OnePlus and BBK Electronics by different dealers in Chennai and Punjab in 2010. Dealer’s Location OnePlus BBK Electronics Chennai a b Punjab c d Chennai e f Punjab g h Chennai i j Punjab k l Chennai m n 1. What is the population standard deviation of sales of OnePlus? Solution: Let $ m_x $ and $ \sigma_x $ be the mean and population standard deviation of sales of OnePlus, respectively.

Profile SVG
Week 10 Graded Assignment Statistics Graded Assignment

There are $2^n$ numbered cards in a deck, among which ${}^n C_i$ cards bear the number $i$ ($i = 0,1,2,\ldots,n$). From the deck, $m$ cards are drawn with replacement. What is the expectation of the sum of their numbers? (Enter the answer correct to one decimal accuracy.)

Profile SVG
Week 11 Graded Assignment Statistics Graded Assignment

A match predictor claims that he can predict the result of a match correctly $x\%$ of the time. It is agreed that his claim will be accepted if he correctly predicts the results of at least $m$ of $n$ matches. What is the probability that his claim gets rejected?

Profile SVG
Week 4 Statistics Graded Assignment Statistics Graded Assignment

Questions (1)–(6): Based on Table 4.1.G Dealer’s Location OnePlus BBK Electronics Chennai 2 | 2 | | Punjab |