CHIRAG GOYAL — Published On April 23, 2021 and Last Modified On June 24th, 2022
This article was published as a part of the Data Science Blogathon.

## Introduction

Statistics is the heart of Machine Learning

Statistical methods are key to find the answers to the questions which we aim to acquire from the data and they are the pillars to all the machine learning approaches.

In this article, I have curated a list of 25 Questions related to Statistics and Probability in Data Science.

## Let’s begin with the Questions,

Section-1

1. Which of the following relation is correct for a negative skewed distribution?

(a) Mean=Mode=Median

(b) Mean>Median>Mode

(c) Mode>Median>Mean

(d) Mean>Mode=Median

Solution:(c)

Explanation: 2. In the symmetric covariance matrix:

(a) Diagonal elements must be positive and other elements are always zero.

(b) Diagonal elements can never be negative and other elements are always positive.

(c) Diagonal elements can never be negative and other elements can be negative or positive.

(d) Diagonal elements can be negative and positive and other elements are always negative.

Solution: (c)

Explanation: In a covariance matrix, the diagonal entries represent covariance of the variable with itself which is equal to the variance of that variable and is calculated as the square of standard deviation. Since variance is always positive, therefore diagonal entries are always positive.

3. Presence of Outliers in a dataset not affects:

(a) Standard deviation

(b) Range

(c) Mean

(d) Inter-quartile Range(IQR)

Solution: (d)

Explanation: The IQR is essentially the range of the middle 50% of the data. Since it uses the middle 50%, therefore it is not affected by the outliers.

4. If X and Y are independent random variables, then which of the following is TRUE?

(a) E(XY)=E(X)E(Y)                   [ E represents Expectation value ]

(b) Cov(X,Y)=0                                     [ Cov represents covariance between variables ]

(c) Var(X+Y)=Var(X)+Var(Y)                              [ Var represents variance ]

(d) All of the above

### Solution: (d)

Explanation: If X and Y are independent then Cov(X,Y)=0 and Var(X+Y) = Var(X)+Var(Y) (∵ 2Cov(X, Y) = 0)

5. For a normal distribution Z, which option is TRUE?

(a) Coefficient of skewness (E(Z3))=0

(b) E(Z)=0 ; E(Z2)=Var(Z)=1

(c) Kurtosis (E(Z4))=3

(d) Its density is symmetric about the mean.

Solution: (d)

Explanation: 6. Let X and Y be normal random variables with their respective means 3 and 4 and variances 9 and 16, then 2X-Y will have normal distribution with parameters:

(a) Mean=2 and Variance=52

(b) Mean=0 and Variance=1

(c) Mean=2 and Variance=1

(d) None of the above

Solution: (d)

Hint: Var(aX + bY) = a2 Var(X) + b2 Var(Y) + 2abCov(X,Y)

7. Suppose X and Y take values {0,1} and are independent with P(X=1)=1/2 and P(Y=1)=1/3. What is the probability that P(X+Y=1)?

(a) 5/18

(b) 1/2

(c) 5/6

(d) 1/6

Solution:(b)

Explanation: P(X +Y =1) = P(X=0).P(Y=1) + P(X=1).P(Y=0) = (1/2)(1/3) + (1/2)(2/3) = 1/2.

8. Let X and Y are random variables with E(X)=μ/2 and E(Y)=μ, then which one is TRUE?

(a) g=X+Y is an unbiased estimator of μ

(b) g = X+Y  is a biased estimator of μ with bias equals to μ

(c) h=X+(Y/2) is an unbiased estimator of μ

(d) h= X+(Y/2) is a biased estimator of μ with bias equals to μ/2

Solution: (c)

Explanation:  E(g)= E(X+Y)= E(X) + E(Y)=3μ/2 ; Bias(g)= E(g)-μ = μ/2
E(h)= E(X+(Y/2))= E(X) + 1/2E(Y) = μ, Bias(h)= E(h)-μ = 0

9. Suppose that X takes values between 0 and 1 and has probability density function(PDF) 2x, then the value of Variance of X2 is :

(a)1/12

(b) 1/18

(c) 1/6

(d) 5/18

Solution:(a)

Hint: Use Var(X2)= E(X4) -(E(X2))2

10. For random variables X and Y, we have Var(X)=1, Var(Y)=4, and Var(2X-3Y)=34, then the correlation between X and Y is:

(a) 1/2

(b) 1/4

(c) 1/3

(d) None of the above

Solution:(b)

Explanation: Var(2X-3Y) = 34

= 4Var(X)+9Var(Y)-12Cov(X, Y)

= 4(1)+9(4)-12Cov(X, Y) = 34

Cov(X, Y)=1/2

11. A fair die is rolled repeatedly until a number larger than 4 is observed. If K is the total number of times that the die is rolled, then P(K=4) is equal to:

(a) 16/81

(b) 8/81

(c) 8/27

(d) 16/27

Solution: (b)

Explanation: P(K=4) = (P(#less than 4 or equal))3.P({4}) = (2/3)3.(1/3) = 8/81.

12. Let X and Y be independent uniform (0, 1) random variables. Define A=X+Y and B=X-Y. Then,

(a) A and B are independent random variables

(b) A and B are uncorrelated random variables

(c) A and B are both uniforms (0,1) random variables.

(d) None of these

Solution: (b)

Explanation: Cov(X+Y, X-Y) = Cov(X, X) – Cov(X, Y) + Cov(Y, X) – Cov(Y ,Y) ⇒ Var(X) – Var(Y) = 0

13. If g is a point estimator of X, then Mean Square error(MSE) for g is:

(a) Variance(g) + Bias(g)

(b) Variance(g) + Bias(g2)

(c) Variance(g) + (Bias(g))2

(d) Variance(g2) + Bias(g)

Solution: (c)

Explanation: MSE(g) = E[ (g-X)2 ] = Var(g-X)  + (E[ g-X ])2 = Var(g) + (Bias(g))2

14. Let X and Y be two random variables and let a, b, c, d be real numbers, then which one of the following is FALSE?

(a) Cov(X+b, Y+d) = Cov(X, Y)

(b) Cov(aX, cY) = ac*Cov(X, Y)

(c) Cov(aX+b, cY+d) = ac*Cov(X, Y)

(d) Corr(aX+b, cY+d) = ac*Corr(X, Y) for a,c>0

Solution: (d)

Explanation: Corr(aX+b, cY+d) = Corr(X, Y)

15. Let X and Y be jointly(bivariate) normal with Var(X) = Var(Y), then:

(a) X+Y and X-Y are jointly normal

(b) X+Y and X-Y are uncorrelated

(c) X+Y and X-Y are independent

(d) All of the above

Solution: (d)

Explanation: If X and Y be the bivariate normal distribution, then any linear combination of X and Y is also normally distributed.

16. Let X1, X2, X3, ——-, Xn be a random sample from a distribution with E(Xi)=μ and Var(Xi)=.σ2 Now, consider two estimators:

g1=X1                       g2=X’=(X1+X2+X3+————-Xn)/n

Which of these estimator has high mean squared error(MSE)?

(a) g1

(b) g2

(c) Same for both g1 and g2

(d) None of the above

Solution: (a)

Explanation: MSE(g1)=E[(g1μ)2] = E[(X1-E(X1))2] = Var(X1) = σ2

MSE(g2)=E[(g2μ)2]= E[(X’-μ)2] = Var(X’-μ) + (E[X’-μ])2 = Var(X’) = σ2/n

17. A random sample of n=6 taken from the population has the elements 6, 10, 13, 14 ,18, 20. Then, which option is False?

(a) Point estimate for population mean is 13.5

(b) Point estimate for population standard deviation is 4.68

(c) Point estimate for population standard deviation is 3.5

(d) Point estimate for standard error of mean is 1.91

Solution: (c)

Explanation: Population mean(X’) = (Σ Xi/n ) = 13.5

Population standard deviation(S) = sqrt( (Σ Xi2/n) – (Σ Xi/n)2 ) = 4.68

Standard error of mean = S/sqrt(n) = 4.68/sqrt(6) = 1.91

Section-2

18. True or False: If the Pearson’s correlation between 2 variables is zero, then they are necessarily independent.

Solution: False

Explanation: Correlation is a measure of linear dependence between the variables.

19. True or False: Let g be an unbiased estimator of X and U be a random variable with zero means, then h=g+U is also unbiased for X.

Solution: True.

Explanation: E(h) =E(g) + E(U) = 0+0 =0( ∵ E(g)=0 due to unbiased estimator)

20. True or False: Let X and Y be two independent standard normal random variables and T=XY2+X+1 and P=X-3, then Cov(T, P)=1

Solution: False.

Hint: Use properties mentioned in Question-14.

21. True or False: Let X has a normal distribution with parameters μ and σ2, then X2 follows a chi-square distribution with parameter 1.

Solution: False.

Explanation: For the given statement to be True, X should be Standard normal distribution(μ=0, σ2=1)

22. True or False: If the characteristic function of a random variable exists, then its expectation and variance will also exist.

Solution: False.

Hint: Moment Generating Function(MGF)

23. True or False: Let X has uniform distribution U(a, b) such that E(X)=2 and Var(X)=3/4, then P(X<1)=1/6.

Solution: True.

Explanation: E(X) = (a+b)/2 = a+b=4   ;      Var(X) = (b-a)2/12 = (b-a)=3  ⇒ X~ U(0.5, 3.5)

24. True or False: The correlation coefficient between X+Y and X-Y, where X and Y are independent random variables with variances 36 and 16 respectively is 6/13.

Solution: False.

Explanation: Corr(X+Y, X-Y) = Cov(X+Y, X-Y)/ Std(X+ Y).Std(X-Y)   [Std= Standard Deviation]

25. True or False: In interval estimation, As the confidence level increases the margin of error decreases.

Solution: False.

Explanation: The Confidence Interval is defined as X ± Z(s/√n)

## End Notes

I hope you enjoyed the questions and were able to test your knowledge about Statistics and Probability in Data Science.

If you liked this and want to know more, go visit my other articles on Data Science and Machine Learning by clicking on the Link

Something not mentioned or want to share your thoughts? Feel free to comment below And I’ll get back to you.

## Chirag Goyal

Currently, I pursuing my Bachelor of Technology (B.Tech) in Computer Science and Engineering from the Indian Institute of Technology Jodhpur(IITJ). I am very enthusiastic about Machine learning, Deep Learning, and Artificial Intelligence.

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