Discrete Probability Distributions Multiple Choice Questions and answers (mcqs)

 1- All Trials are Independent in: ?

(A) Binomial Experiment (B) Hypergeometric Experiment

Option (A) is correct.

Note: But, all Trials are Dependent in Hypergeometric Experiment.

2- In Binomial Experiment, drawing is done: ?

(A) With Replacement (B) Without Replacement

Option (A) is correct.

Note: But, In Hypergeometric Experiment, drawing is done Without Replacement.

3- Binomial Distribution has range from: ?

(A) 0,1,2,.....infinity (B) 0,1,2,......n

Option (B) is correct.

Note: Must remember that also Hypergeometric Distribution has range from 0,1,2,......n.

4- The Probability of failure 'q' is equal to: ?

(A) 1- p (B) 1-q

Option (A) is correct.

5- The Variance of Binomial Distribution is: ?

(A) npq (B) np

Option (A) is correct.

Note: But, In Binomial Distribution, Mean = np.

6- For Negative Binomial Distribution, Mean and Variance have a relation: ?

(A) Mean > Variance (B) Mean < Variance 

Option (B) is correct.

Note: Must focus on that's, In Binomial Distribution, always Mean > Variance.

7- The Mean of Hypergeometric Probability Distribution is: ?

(A) nk/N (B) npq

Option (A) is correct.

8- In Hypergeometric Probability Distribution, Probability 'p' is equal to : ?

(A) k/N   (B) k/n

Option (A) is correct.

Note: Attention!!

Now, I am going to tell you most Important points which nobody tells you about that. Must remember that In sense of Problems solving, In case of With Replacement , always use the formula of Binomial Distribution. But, In case of Without Replacement , always use the formula of Hypergeometric Distribution.

9- If 'n' becomes very large and the probability of success "p" is very small, then that's type of Distribution is defined as: ?

(A) Normal Distribution (B) Poisson Distribution

Option (B) is correct.

Note : For example, in problems solving,, n=500 and p=0.01. then we always use Poisson Distribution.

10- In Poisson Distribution, Mean and Variance have a relation: ?

(A) Mean = Variance (B) Mean < Variance

Option (A) is correct.

Note: Must remember that In Poisson Distribution, Mean = np and Variance = np.

11- The Product of N and P(X) ; NP(X) is equal to: ?

(A) Expected Frequency (B) Observed Frequency

Option (A) is correct.

12- In Poisson Process, Average number of occurances per unit time is equal to: ?

(A) Lemda (B) t

Option (A) is correct.

Note: t shows 'time interval".

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Random Variables Multiple Choice Questions and answers (mcqs)

 1- Number of heads when when we toss some coins is an example of: ?

(A) Discrete Random Variable (B) Continuous Random Variables

Option (A) is correct.

Note: Must remember that Discrete Random Variable has Countable Number of Values. But, Continuous Random Variable has Uncountable Number of Values. Here, Option (A) is correct because we can count "Number of Heads when we toss 2 or more coins.

2- Lengths, Widths is an example of: ?

(A) Discrete Random Variable (B) Continuous Random Variable

Option (B) is correct.

3- A variable which can assume values of random experiment is called as:

(A) Random Variable (B) Random Experiment

Option (A) is correct.

4- Probability Distribution is also called as:

(A) Distribution Function (B) Probability Mass Function

Option (B) is correct.

Note: Q- What is Probability Distribution?

An arrangement of probabilities against each possible value of discrete random variable is called as Probability Distribution.

5- Distribution Function , F(x) =: ?

(A) P(X<x) (B) f(x)

Option (A) is correct.

Note: In reality,  P(X<x) that is; Probability of ( X less than and equal to x).

6- Distribution Function is also defined as: ?

(A) Cumulative Probability Distribution (B) Probability Mass Function

Option (A) is correct.

7- Probability Density Function is always: ?

(A) Discrete (B) Continuous

Option (B) is correct.

Note: Probability Density function is denoted as " f(x)" .

8- Expectation E(X) = ?

(A) Sum of X.P(X) (B) Nothing

Option (A) is correct.

9- E(aX + b) =: ?

(A) aE(X) + b (B) E(x) + b

Option (A) is correct.

Note: Must remember that Most Important point which nobody tells you about that, " Mean = Expectation ". Next, Mean of "a" = E(a) = a. that is; Expectation of a constant value is equal to constant itself.

10- Probability Distribution for only two variables at a time is called as: ?

(A) Bi-Variate Probability Distribution (B) Joint Probability Distribution

Option (A) is correct.

Note: But, Probability Distribution for two or more than two variables is called as Joint Probability Distribution. Must focus on that point " Bi-Variate Probability Distribution = Joint Probability Distribution because both distributions have two variables. When we check that it has three variables then it must be purely 'Joint Probability Distribution' .

11- Two random variables X and Y will be Independent If: ?

(A) f(X,Y) = g(X).h(Y)  (B) f(X,Y) = f(X)

Option (A) is correct.

Note: But, If two random variables X and Y will be Dependent then " f(X,Y) does not equal to g(X).h(Y).

12- Covariance of X and Y ; Cov(X,Y) and Correlation Coefficient are zero, When 

(A) X and Y are Independent (B) X and Y are Dependent

Option (A) is correct.

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Probability Multiple Choice Questions and answers (mcqs)

 1- Any particular performance of random experiment is called as:

(A) Trial (B) Sample Space

Option (A) is correct.

2- In S = { 1,2,3,4,5,6} ;  1,2,3,4,5,6 are:

(A) Outcomes (B) Events

Option (A) is correct.

Note: 1,2,3,4,5,6 are the results of random experiment which are called outcomes.

3- Simple Event always have _ outcomes:?

(A) triple outcomes (B) single outcome

Option (B) is correct.

Note: Suppose, A = {1} which have a single outcome is defined as simple event. But, If we suppose, A = {1,2} then set A is defined as Compound event because it has more than 1 outcomes.

4- Exhaustive Events is defined as:

(A) A U B = S (B) AB = S

Option (A) is correct.

Note: The Union of two events A & B is equal to Sample Space 'S' is defined as Exhaustive Events.

5- The Sum of Probability never exceeds:

(A) 2 (B) 1

Option (B) is correct.

6- Given A = { 1,2 }, B = { 4,5,6 } ; A and B are called:

(A) Equally Likely Events (B) Mutually Exclusive Events

Option (B) is correct.

Note: Let's suppose, A = {1,2} , B = {4,5,6}. You can check that both have no common points. So, must remember that The Events A and B which have no common points are called as Mutually Exclusive Events.

But, Let's suppose , A = {2,4,5} and B ={4,5,6} ; Both have common points. So, must remember that The Events which have common points are called as Equally Likely Events.

7- Probability of Sample Space; P(S) = : ?

(A) 1 (B) 0

Option (A) is correct.

Note: Suppose that A = {4,6,2,3,1,5}, For one Dice, S = {1,2,3,4,5,6}, P(S) = number in A/number in Sample Space , that is; P(S) = n(A)/n(S)= 6/6 = 1.

8- Given ; A = {7,8,9} , S = {1,2,3,4,5,6}; P(A) =: ?

(A) 2/3 (B) 1/2

Option (B) is correct.

Note: P(A) = n(A)/n(S) = 3/6 = 1/2.

9- If A and B are independent then P(A intersection B) =: ?

(A) P(A) . P(B) (B) P(A) + P(B)

Option (A) is correct.

10- In Independent Events, drawing is done: ?

(A) Without Replacement (B) With Replacement 

Option (B) is correct.

Note: But, In Dependent Events, drawing is done Without Replacement.

11- Permutation is always in:

(A) Ordered (B) Disordered

Option (A) is correct.

Note: But, Combination is Disordered.

12- If A and B are Mutually Exclusive , then P(A U B)=: ?

(A) P(A) + P(B) (B) P(A) . P(B)

Option (A) is correct.

13- Poker hand shows:

(A) cards (B) Drawn Cards

Option (B) is correct.

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Index Numbers Multiple Choice Questions and answers (mcqs)

 1- The formula of  link relatives is:

(A) Pn/Pn-1 * 100 (B) Pn/Po * 100

Option (A) is correct.

Note: Formula of Price relative is : Pn/Po * 100.

Formula of Quantity relative is : Qn/Qo * 100.

Pn is the price of current year & Po is the price of base year.

2- In Fix-Base Method, we calculate:

(A) Price Relatives (B) Quantity Relatives

Option (A) is correct.

Note: But, In Chain-Base Method , we calculate Link Relatives and then make chain indices.

3- Index number for base period is always taken as:

(A) 200 (B) 100

Option (B) is correct.

4- The formula of Fisher's 'Ideal' Price Index is:

(A) Pon = Square Root of L * P (B) Pon = L * P

Option (A) is correct.

Note: L shows Laspeyre's Price Index and P shows Paasche's Price Index.

5- Value Relative =:

(A) Price Relative * Quantity Relative (B) Nothing

Option (A) is correct.

6- In Time-Reversal Test, Pon * Pno =:

(A) 2 (B) 1

Option (B) is correct.

7- In Tests of Index numbers , we neglect the factor:?

(A) 100 (B) 300

Option (A) is correct.

8- In Factor- Reversal Test, Price Index * Quantity Index = :?

(A) Weighted Index (B) Value Index

Option (B) is correct.

9- Only Fisher's Index satisfies:

(A) Factor Reversal Test (B) Circular Test

Option (A) is correct.

10- In Circular Test, Pab * Pbc * Pca = :?

(A) 100 (B) 1

Option (B) is correct.

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More Multiple Choice Questions and answers are available here!

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Measures of Dispersion, Moments and Skewness Multiple Choice Questions and answers (mcqs)

 1- Deviation from Average is called as:

(A) Dispersion (B) Variance

Option (A) is correct.

2- The Coefficient of Mean Deviation is:

(A) Mean Deviation/Average (B) Average

Option (A) is correct.

3- The Mean of squared deviation from mean is defined as:

(A) Standard Deviation (B) Variance

Option (B) is correct.

4- If data is more consistent then co-efficient of variation will be:

(A) Least (B) Maximum

Option (A) is correct.

5- Given data:

X: 4,4,4,4

Variance=?

(A) 1 (B) 0

Option (B) is correct.

6- Var(x+a)= ?

(A) Var(x) (B) Var(x-a)

Option (A) is correct.

Note: Var(x-a) = Var(x)

7- Given data: Var(x)=2, Var(y)=4 then find Var(x - y)= ?

(A) 7 (B) 6

Option (B) is correct.

8- Mean of Deviation to which power are raised are called as:

(A) Moments (B) Mean Deviation

Option (A) is correct.

9- If Mean > Median > Mode, then distribution is:

(A) Positively Skewed (B) Negatively Skewed

Option (A) is correct.

Note : If Mean < Median < Mode then distribution is Negatively Skewed.

10- If 3rd Moment about Mean (m3) = 0 , then distribution is:

(A) Skewed (B) Symmetrical 

Option (B) is correct.

11- The degree of peakdness is defined as:

(A) Kurtosis (B) Moment

Option (A) is correct.

12- The Standard value of Kurtosis is:

(A) 9 (B) 3 

Option (B) is correct.

13- Second moment about mean (m2) =:

(A) Variance (B) Standard Deviation 

Option (A) is correct.

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Measures of Central Tendency or Averages Multiple Choice Questions and answers (mcqs)

 1- The Central value of data is called as:

(A) Average (B) Mean

Option (A) is correct.

2- In Average, Frequency 'f' always come in:

(A) Group Data (B) Ungroup Data

Option (A) is correct.

3- Summation a =:

(A) mb (B) na

Option (B) is correct.

4- Sum of deviation from mean =:

(A) 0 (B) 1

Option (A) is correct.

5- Sum of squared deviation from mean is always:

(A) maximum (B) minimum

Option (B) is correct.

6- Geometric Mean can never be calculated If any value in the data is:

(A) Negative or Zero (B) Five or Six

Option (A) is correct.

7- Harmonic Mean is used when:

(A) speed type of data is given (B) percentage increase or decrease types of data is given

Option (A) is correct.

Note: Geometric Mean is used when percentage increase or decrease types of data is given.

8- Reciprocal of Mean of Reciprocal values is called as:

(A) Geometric Mean (B) Harmonic Mean

Option (B) is correct.

9- A value which divides a data set into two equal parts is called as:

(A) Median (B) Mode

Option (A) is correct.

Note: Most repeated value in the data is called as Mode.

10- Given data;

X: 8,4,3,2,5

Median =?

(A) 3 (B) 4

Option (B) is correct.

11- If 'a' and 'b' are two numbers then Geometric Mean will be:

(A) Square root of 'ab' (B) Square of 'ab'

Option (A) is correct.

12- Emperical formula of Mean, Median & Mode is:

(A) Mode = 3 Median - 2 Mean

(B) Mode = 3 Mean - 2 Median

Option (A) is correct.

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Presentation of Data Multiple Choice Questions and answers (mcqs)

 1- Limits of Discrete Groups are called:

(A) Class Limits (B) Class Boundaries

Option(A) is correct.

2- The difference of Maximum value and Minimum value is called as:

(A) Range (B) Ogive 

Option (A) is correct.

3- The frequency less than the upper class boundary of a group is called:

(A) Class boundary (B) Cumulative frequency

Option (B) is correct.

4- Smoothness of frequency polygon is said to be:

(A) frequency curve (B) curve fitting

Option (A) is correct.

5- Ogive is also called as:

(A) Frequency Polygon (B) Cumulative Frequency Polygon

Option (B) is correct.

6- The difference between upper class boundaries and lower class boundaries of each group is called as:

(A) Class Interval (B) Range

Option (A) is correct.

7- Mid point is also said to be:

(A) Class Mark (B) Class Boundary

Option (A) is correct.

8- Graph of adjacent rectangles in which class boundaries are taken along x-axis and frequencies along y-axis is called as:

(A) Frequency Polygon (B) Histogram 

Option (B) is correct.

9- Pie-chart has a shape like:

(A) Pizza (B) Burger

Option (A) is correct.

10- In Stem-and-leaf Display, leaf is the:

(A) rest of the numbers (B) first or leading digits of number

Option (A) is correct.

11- If class interval is not given then we have to make;

(A) Steps of frequency polygon (B) Steps of frequency distribution

Option (B) is correct.

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Next post will be: Measures of Central Tendency or Averages mcqs (Multiple Choice Questions)

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Introduction to Statistics Multiple Choice Questions and answers (mcqs)

1- The word 'STATISTICS' refers to:

(A) numerical facts systematically arranged (B) Nothing

Option (A) is correct.

2- Population is a:

(A) Statistic (B) Parameter

Option (B) is correct.

3- A cricket player wants to find his score average for the last 20 games is an example of:

(A) Descriptive Statistics (B) Inferential Statistics

Option (A) is correct.

4- The age of a person is an example of:

(A) Continuous variable (B) Discrete variable

Option (A) is correct.

5- The example of Nominal Scale is:

(A) Objects (B) Gender (Male & Female)

Option (B) is correct.

6- Rating students in Marks as 'poor ,good, excellent' is an example of:

(A) Ordinal Scale (B) Internal Scale

Option (A) is correct.

7- Collection of Statistical Information is called:

(A) Data (B) Group

Option (A) is correct.

8- The data which has undergone any statistical treatment is called:

(A) Primary data (B) Secondary data

Option (B) is correct.

9- Qualitative Variable is also called:

(A) Attribute (B) Sample

Option (A) is correct.

10- Age and Weight is an example of:

(A) Qualitative variable (B) Quantitative variable

Option (B) is correct.

11- Sample is part of:

(A) Population (B) Statistic

Option (A) is correct.

12- The number of children in a family is an example of :

(A) Discrete Variable (B) Continuous Variable

Option (A) is correct.

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Statistics Definition & Examples in Real Life

 STATISTICS : Statistics is the science of collection, interpretation, observation, analysis and presentation of numerical facts.

REFERS : The word Statistics refers to " numerically facts systematically arranged ".

DEALINGS : Statistics deals with variability and uncertainity.

EXAMPLES OF STATISTICS IN REAL LIFE : 

There are many examples of statistics in real life. But, I will tell here three top examples.

1- Survey

For example, we can say that Housing Survey, COVID-19 Survey etc,

2- Division

For example, when we divide the whole population into sub-populations. Let's take Pakistan, we divide it into sub-populations ' Lahore, Islamabad, Sialkot, Karachi etc,. You can also suppose America, we divide it into U.K, USA, France etc,. 

3- Lucky Draws

For example, lottery, lucky draws ,that is , we get a big amount money free of cost.

We mostly use STATISTICS!

I hope that you got a lot of points and concepts.

Thank you

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