wsw10-Graded_Assignment_Sep_2025_Statistics_I_Qualifier (1)

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

The due date for submitting this assignment has passed. Due on 2025-12-03, 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: 2025-12-03, 12:50 IST

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Question 1

There are $2^7$numbered cards in a deck among which ${}^7{C}_i$ cards bear the number $i;i=0,1,2,...,7$. From the deck, 11 cards are drawn with replacement. What is the expectation of the sum of their numbers?(Enter the answer correct to 1 decimal accuracy) Your Answer: 38.5

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

Accepted Answers:

(Type: Range) 38.4,38.6

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Question 2

An unbiased die is thrown 15 times. After each throw a ’+’ is recorded for 2 or 5 and ’-’ is recorded for 1,3,4 or 6, the signs forming an ordered sequence. To each, except the first and last sign, a random variable $X_i;i=1,2,...,13$ is associated which takes the value $1$ if both of its neighbouring sign differs from the one between them and $0$ otherwise. If the random variable $Y$ is defined as $Y=3S+35$ where, $S={\sum }_{i=1}^{13}{X}_i$.
Find the expected value of $Y$.

• $\frac{78}{9}$
• $\frac{341}{9}$
• $\frac{26}{9}$
• $\frac{393}{9}$

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

Accepted Answers:

• $\frac{393}{9}$

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Question 3

An unbiased die is thrown 15 times. After each throw a ’+’ is recorded for 2 or 5 and ’-’ is recorded for 1,3,4 or 6, the signs forming an ordered sequence. To each, except the first and last sign, a random variable $X_i;i=1,2,...,13$ is associated which takes the value $1$ if both of its neighbouring sign differs from the one between them and $0$ otherwise. If the random variable $Y$ is defined as $Y=3S+35$ where, $S={\sum }_{i=1}^{13}{X}_i$.
Which of the following statement(s) is/are true?

• $E(Y)=9E(S)+35$
• $V(Y)=9V(S)$
• $V(Y)=9V(S)+35$
• $E(Y)=3E(S)+35$

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

Accepted Answers:

• $V(Y)=9V(S)$
• $E(Y)=3E(S)+35$

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Question 4

Amandeep is in the middle of a bridge of infinite length. He takes the unit step to the right with probability 0.66 and to the left with probability 0.34. Assume that the movements are independent of each other.
Hint: Consider the random variable $X_i$ associated with the $i^{th}$ step defined as:
$X_i$=$\{\begin{array}{ll}1& \text{if the step of Amandeep is towards the right }\\ -1& \text{if the step of Amandeep is towards the left }\end{array}$
What is the expected distance between the starting point and end point of Amandeep after 7 steps?

• $0.32$
• $-2.24$
• $2.24$
• $-0.32$

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

Accepted Answers:

• $2.24$

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Question 5

Amandeep is in the middle of a bridge of infinite length. He takes the unit step to the right with probability 0.66 and to the left with probability 0.34. Assume that the movements are independent of each other.
Hint: Consider the random variable $X_i$ associated with the $i^{th}$ step defined as:
$X_i$=$\{\begin{array}{ll}1& \text{if the step of Amandeep is towards the right }\\ -1& \text{if the step of Amandeep is towards the left }\end{array}$
What is the variance distance between the starting point and end point of Amandeep after 7 steps?

• $0.9$
• $4.02$
• $1$
• $6.28$

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

Accepted Answers:

• $6.28$

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Question 6

A box contains 6 white and 7 black balls. 3 balls are drawn at random without replacement. Find the expected value of the number of white balls drawn. (Enter the answer correct to 2 decimal places) Your Answer: 1.38

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

Accepted Answers:

(Type: Range) 1.35,1.41

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Question 7

Rohit wants to open his door with $5$ keys(out of which $1$ will open the door) and tries the keys independently and at random. If unsuccessful keys are eliminated from further selection, then Find the expected number of trials required to open the door.
Hint: Suppose Rohit gets the first success at $x^{th}$ trial, i.e., he is unable to open the door in the first $(x-1)$ trials.
And, P(he gets first success at second trial)=$(1-\frac{1}{5})\times \frac{1}{4}$

• 1
• 9
• 3
• 2

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

Accepted Answers:

• 3

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Question 8

$X$ and $Y$ are independent random variables with means 10 and 17, and variances 5 and 5 respectively. Find the variance of $6X+3Y$. Your Answer: 225

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

Accepted Answers:

(Type: Numeric) 225

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Question 9

Table Q10.1.G. represents the probability mass function of a random variable X

$X$

-2

2

5

$P(X=x)$

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{5}{12}$

Table Q10.1.G: Probability Mass Function

Calculate the value of $E(2X+1{)}^2$. (Enter the answer correct to 2 decimal places) Your Answer: 61

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

Accepted Answers:

(Type: Range) 60.97,61.03

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Question 10

Suppose that $X$ is a random variable for which $E(X)=12$ and $Var(X)=25$. Find the positive values of $a$ and $b$ such that $Y=aX-b$, has expectation $0$ and variance $1$.

• $\frac{1}{\sqrt{25}}$ , $\frac{12}{\sqrt{25}}$
• $\frac{1}{25}$ , $\frac{12}{\sqrt{25}}$
• $\frac{-1}{\sqrt{25}}$ , $\frac{-12}{\sqrt{25}}$
• $1$ , $12$

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

Accepted Answers:

• $\frac{1}{\sqrt{25}}$ , $\frac{12}{\sqrt{25}}$

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