wsq3-Graded_Assignment_Sep_2025_Statistics_I_Qualifier (4)

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

Due date for this assignment: 2026-12-18, 20:00 IST. You may submit any number of times before the due date. The final submission will be considered for grading.

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Question 1

An entrepreneur has invested in four startups. If the startup makes profit, it will yield respective profits of $20,25,30$ and $40$ (in units of lakh). On the other hand, for each startup the entrepreneur may not gain profit, he will incur a loss of $10$ (in units of lakh). If the probabilities that the entrepreneur will make profits from these startups are $0.1,0.3,0.2,$ and $0.1$ respectively, what is the expected total profit (A negative profit indicates loss)?

• 19.5
• 9.5
• -13.5
• -9.5

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Question 2

If the graph of pmf of a binomial random variable of n = 10 is exactly symmetric about X = 5, then the value of p could be: (More than one option could be correct.)

• 0
• 0.25
• 0.5
• 0.75
• 1

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Question 3

A robber who needs to open a lock of a house has 21 keys out of which only one key can unlock the house. The chance of selecting a key is equally likely for all the keys. The robber will not use the keys that did not open the lock in the upcoming trials. Let $X$ be defined as the number of trials robber requires to open the house. What is the value of $E(X)$? Your Answer: (Not answered)

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Question 4

If the median of the dataset $x_i$, where $i=1,2,3,\cdots ,n$, is 15, what is the median of the dataset $2x_i-2$, where $i=1,2,3,\cdots ,n$? Your Answer: (Not answered)

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Question 5

Let $X$ be a discrete random variable having the following probability mass function:

$P(X=x)=k\times \frac{{}^3{C}_x}{{}^3{C}_{3-x}}$

where $x$ is taking the value 0, 1, and 2. Find the value of $k$. (Enter your answer correct up to 2 decimal accuracy) Your Answer: (Not answered)

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Question 6

Which of the following variables is/are numerical?

• Height (in metres)
• Aadhar card number
• Pan card number
• Passenger Name Reservation (PNR) number
• Stop watch time (in seconds)
• Pin code

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Question 7

The variance of a binomial random variable $X$ is plotted against varying values of $p$ with $n$ kept constant. It is observed that the maximum value of variance is $40$, then what is the value of $n$?

• 80
• 160
• 40
• 400

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Question 8

John parked his car among 10 cars in a row in a mall parking, not at either end, which has a capacity of 11 cars. In how many ways is it possible that on his return from the mall, he finds that exactly 5 (including his car) of the 11 places are still occupied, if both neighbouring parking plots around his car are empty?

• 126
• 462
• 70
• 35

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Question 9

Two players, Sam and Nina, throw a pair of dice alternatively and independently. Sam wins if he throws a sum of six points before Nina throws a sum of seven points, while Nina wins if she throws a sum of seven points before Sam throws a sum of six points. If Nina begins the game, then what is the probability of Nina winning the game? (Enter the answer correct to two decimal places) Your Answer: (Not answered)

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Question 10

Suppose that the random variable $X$ assumes three values $0$, $1$ and $2$ with probabilities $\frac{1}{3}$, $\frac{1}{6}$ and $\frac{1}{2}$ respectively. Calculate the value of $F(1)$. Your Answer: (Not answered)

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Question 11

Let $X\sim Exp(\lambda )$. Then find the value of $z$ such that $P(X\ge 100|X\ge 50)=P(X\ge z)$.

• 100
• 150
• 50
• 49

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Question 12

According to a survey, the time taken by teenagers to finish a box of donuts is uniformly distributed between [1,6] minutes. A café in New Delhi used this data for its advertisement that challenges teenagers to finish a box of donuts within 100 seconds to win a grand prize. A teenager, Ajay, accepted the café’s challenge. What is the probability that he will finish the box of donuts within 4 minutes, given that he lost the challenge?

• $\frac{4}{5}$
• $\frac{1}{2}$
• $\frac{7}{13}$
• $\frac{3}{5}$

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Question 13

A company has predicted that the next year’s profit will follow the probability distribution shown in Table 1.E.2. The random variable $X$ denote the profit in million rupees. Loss is denoted by a negative profit.

But the company cannot retain all the profits to itself since it has to share 15% of the profit to its investors. The amount that the company retains is given by $Y$ = 0.85X. What is the approximate standard deviation (in million rupees) of $Y$ .

• 0.17
• 1.06
• 0.19
• 0.75

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Question 14

The duration of a video game is a random variable with an Exponential distribution with mean $20$ minutes. Suppose that when you enter a video arcade, two players are playing the game (independently). Find the probability that exactly one of the two will be playing for less than $40$ minutes?

• $e^{-2}(1-e^{-2})$
• $2e^{-2}(1-e^{-2})$
• $e^{-8}(1-e^{-8})$
• $2e^{-8}(1-e^{-8})$

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Question 15

A fruit seller has $6$ dozen bananas out of which $12$ are rotted. If a customer wants to purchase some bananas and chooses $6$ bananas (without replacement), then how many rotten bananas can expect customer? Your Answer: (Not answered)

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Question 16

The average number of penalties charged by a police in a day for not following the traffic rules is 5. Assume that the number of penalties charged on different days are independent and follows a Poisson distribution. Find the probability that less than 4 penalties will be charged in a day. (Enter the answer correct to 2 decimal accuracy.) Your Answer: (Not answered)

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