wsw12-Graded_Assignment_Sep_2025_Statistics_I_Qualifier (3)

Graded Assignment - (Sep 2025 - Statistics I - Qualifier)

The due date for submitting this assignment has passed. Due on 2025-12-10, 23:59 IST. You may submit any number of times before the due date. The final submission will be considered for grading.

Last Submitted: You have last submitted on: 2025-12-10, 07:18 IST

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Question 1

Let a random variable is uniformly distributed over $[a,b]$ with expectation and variance $8.0$ and $\frac{4}{12}$ respectively. Find the value of $ab$. Your Answer: 63

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

(Type: Numeric) 63

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Question 2

Probability density function of a random variable $X$ is given by:
$f(x)$=$\{\begin{array}{ll}k{x}^2& \text{if x ∈ [0,1]}\\ 0& \text{otherwise}\end{array}$
Find $P(0.3<X<0.8)$?
Hint: ${\int }_a^{b}c{x}^{2}dx=c\frac{b^3-a^3}{3}$ where, $c$ is a constant.

• $\frac{0.8^2-0.3^2}{9}$
• $0.8^3-0.3^3$
• $0.8^2-0.3^2$
• $\frac{0.8^3-0.3^3}{9}$

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

• $0.8^3-0.3^3$

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Question 3

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 $114$ litres given that on that day, more than $108$ litres of milk was produced? (Enter the answer correct to 2 decimal place) Your Answer: 0.50

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

(Type: Range) 0.47,0.53

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Question 4

The time that Jan shatabdi express will reach the Delhi station is uniformly distributed between $3$ p.m and $8$ p.m. What is the probability that the train reaches Delhi exactly at $4$ p.m? Your Answer: 0

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

(Type: Numeric) 0

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Question 5

The length of time for one person to be served at a restaurant is exponentially distributed with an expected time of $4$ minutes. If $5$ persons arrived at the restaurant then what is the probability that at least $2$ of them will be served in more than $6$ minutes? (Assume that each person is served independently)

• $1-{\left(1-e^{-1.5}\right)}^5-\left[5e^{-1.5}{\left(1-e^{-1.5}\right)}^4\right]$
• $1-{\left(e^{-1.5}\right)}^5-\left[5{\left(e^{-1.5}\right)}^4\left(1-e^{-1.5}\right)\right]$
• $1-{\left(e^{-24}\right)}^5-\left[5{\left(e^{-24}\right)}^4(1-e^{-24})\right]$
• $1-{\left(1-e^{-24}\right)}^5-\left[5e^{-24}{\left(1-e^{-24}\right)}^4\right]$

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

• $1-{\left(1-e^{-1.5}\right)}^5-\left[5e^{-1.5}{\left(1-e^{-1.5}\right)}^4\right]$

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Question 6

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 \leq3}\\ 0.4 & \text{if 3 \leq x \leq 4}\\ 0 & \text{otherwise} \end{cases}

Find the probability that $0<x<2.5$. (Answer upto three decimal points)

Hint : Try to find the area under pdf using integration, otherwise draw the graph of pdf of $X$. Your Answer: 0.425

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

(Type: Range) 0.421, 0.429

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Question 7

In a certain exhibition, the time for the next visitor to come is exponentially distributed with a standard deviation of $11$ minutes. What will be the expected time (in minutes) for two visitors to arrive if one comes after the other? (Note that the arrival of a visitor is independent of the arrival of the previous visitor.)

• $\frac{2}{11}$
• 11
• $2\times 11$
• $\frac{1}{11}$

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

• $2\times 11$

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Question 8

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 $43$ minutes is $\frac{1}{6}$. Find the expected time duration (in minutes) of a badminton match. (Enter the answer correct to 1 decimal place) Your Answer: 47.0

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

(Type: Range) 46.7,47.3

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Question 9

The probability density function of the time $X$ (in minutes) between calls at the customer care is given by:
$f(x)$=$\{\begin{array}{ll}\frac{1}{7}{e}^{\frac{-x}{7}}& \text{if x > 0}\\ 0& \text{otherwise}\end{array}$
Find the probability that time between calls exceeds the mean time.

• $e$
• $\frac{1}{e}$
• $1-\frac{1}{e}$
• $1-e$

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

• $\frac{1}{e}$

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Question 10

The lifetime of a light bulb is exponentially distributed with a mean life of $18$ months. If there are $40$% chances that a light bulb will last at most $t$ months, then what is the value of $t$?

• $18\times \ln \left(\frac{100-40}{100}\right)$
• $18\times \ln \left(\frac{100}{100-40}\right)$
• $\frac{1}{18}\ln \left(\frac{100}{100-40}\right)$
• $\frac{1}{18}\ln \left(\frac{100-40}{100}\right)$

Status: Yes, the answer is correct. Score: 1

Accepted Answers:

• $18\times \ln \left(\frac{100}{100-40}\right)$

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