1. Standard Normal Distribution

For a standard normal distribution, what are its mean and variance?

Pick ONE option:

• 0, 1
• 1, 1
• 0, 0.68
• 1, 0.68

2. Probability - II

In a family, the probability that at least one parent falls ill is 20%. If the probability that the father falls ill is 12% and the probability that both the parents fall ill is 5%, what is the probability that the mother falls ill?

Pick ONE option:

• 3%
• 13%
• 10%
• 8%

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3. Sample Mean, Standard Deviation and Median

The following data is from a random sample: 7, 1, 9, 13, 10. Compute the sample mean, sample standard deviation, and sample median.

Pick ONE option:

• 8, sqrt(40), 19
• 7, sqrt(30), 10
• 8, sqrt(25), 13
• 8, sqrt(20), 9

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4. Central Limit Theorem (CLT)

Let X1,X2,...,X75X_1, X_2, ..., X_{75}X1​,X2​,...,X75​ be i.i.d., each with expected value μ = E(X_i) = 2, and variance σ2=Var(Xi)=3\sigma^2 = Var(X_i) = 3σ2=Var(Xi​)=3. Approximate P(X_1 + X_2 + ... + X_75 > 120) using the central limit theorem. For the solution, it can be assumed that P(∣Z∣<2)≈0.95P(|Z| < 2) ≈ 0.95P(∣Z∣<2)≈0.95, where Z is a random variable that has a standard normal distribution.

Pick ONE option:

• 0.95
• 0.85
• 0.975
• 0.6

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5. Compare Scores

A student received the following marks for exams in 2 subjects. Which subject showed better relative performance? (Assume scores follow a normal distribution)

Subject	Marks obtained (out of 100)	Mean Score	Standard Deviation
Maths	70	60	15
Science	72	68	6

Pick ONE option:

• Maths
• Science
• The performance is the same in both subjects.

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6. Statistical Tests

Choose the correct statement(s) about the statistical tests.

Pick ONE OR MORE options:

• Z-test and T-test can only be used if the underlying distribution is normal.
• Chi-square test can be used if the data is not normally distributed.
• T-test requires population mean and standard deviation to be given.
• Z-test can be used to validate a hypothesis that the sample drawn belongs to the same population.

7. Sort an Array

Given an array of integers, any array element can be moved to the end in one move. Determine the minimum number of moves required to sort the array, ascending.

Example:

arr = [5, 1, 3, 2]

• Move the value arr[2] = 3 to the end to get arr = [5, 1, 2, 3].
• Move arr[0] = 5 to the end to achieve the sorted array, arr = [1, 2, 3, 5].
• The minimum number of moves required to sort the array is 2.

Function Description: Complete the function getMinimumMoves in the editor below.

getMinimumMoves has the following parameter:

• int arr[n]: an array of integers

Returns:

• int: the minimum number of moves needed to sort the array in ascending order

Constraints:

• 1 ≤ n ≤ 10^5
• 0 ≤ arr[i] < 10^6
• Array elements are distinct.

8. Disk Space Analysis

A company is performing an analysis on the computers at its main office. The computers are spaced along a single row. For each group of contiguous computers of a certain length (segment), determine the minimum amount of disk space available on a computer. Return the maximum of these values as your answer.

Example:

• n=4n = 4n=4, the number of computers
• space=[8,2,4,6]space = [8, 2, 4, 6]space=[8,2,4,6]
• x=2x = 2x=2, the segment length

The free disk space of computers in each of the segments is [8,2],[2,4],[4,6][8, 2], [2, 4], [4, 6][8,2],[2,4],[4,6]. The minima of the three segments are [2,2,4][2, 2, 4][2,2,4]. The maximum of these is 4.

Function Description: Complete the function segment in the editor below.

segment has the following parameters:

• int x: the segment length to analyze
• int space[n]: the available hard disk space on each of the computers

Returns:

• int: the maximum of the minimum values of available hard disk space found while analyzing the computers in segments of length xxx.

9. Extraordinary Substrings

Each character of the lowercase English alphabet has been mapped to digits as shown in the figure. The numerical value corresponding to each letter is its mapped value.

An extraordinary substring is one whose sum of the mapped values of each letter is divisible by its length. Given string input_str, count its total number of non-empty extraordinary substrings.

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