SUBJECT :- STATISTICS th Standard :- 12 Commerce
Transcript of SUBJECT :- STATISTICS th Standard :- 12 Commerce
SUBJECT :- STATISTICS Standard :- 12th Commerce
Answer the following question in one sentence
1) Which index number is used in India to find the rate of dearness allowance?
Ans: - Cost of living index number is used in India to find the rate of dearness allowance.
2) What is the main limitation of Scatter diagram method?
Ans: - The main limitation of scatter diagram method is that we cannot obtain the exact
measure of strength of correlation.
3) Define Regression coefficient.
Ans: - The co-efficient of π ππ π₯ in the linear regression model π¦ = π + ππ₯ is known
as co-efficient of regression. OR The approximate change in the value of dependent
variable for a unit change in the value of independent variable. h represents the
appropriate increase or decrease in the value of Y per one unit increase or decrease in
the value of X.
4) State the interval required to know the trend in agricultural yield or industrial production.
Ans: - To require the trend in agricultural yield or industrial production. 10-15 years time
interval is required.
5) State the formula for the probability of occurrence of at least one event out of three events
A, B and C.
Ans: - Occurrence at least one event out of three evens A, B and C or π΄ π π΅ π πΆ.
π(π΄ βͺ π΅ βͺ πΆ)
= π(π΄) + π(π΅) + π(πΆ) β π(π΄ β© π΅) β π(β© π) β π(π΅ β© π) + π(π΄
β© π΅ β© π)
6) Define discrete random variable.
Ans: - A random variable X, which is capable of taking any real value of set R or its
subset is called a discrete random variable
7) What is the probability that a continuous random variable takes definite value ?
Ans: -. The probability that a continuous random variable takes definite value is always
'0; (zero)
8) For a normal distribution, the estimated value of quartile deviation is 12. Find the value of
its standard deviation.
Ans: - In normal distribution fourth quartile =2
3π,therefore 12 =
2
3π β΄ π =
12Γ3
2= 18
Thus, the value of standard deviation is 18.
9) Explain the meaning of x.
Ans: -. If by decreasing the positive value of a variable X or by increasing negative value
of the variable X ; the value of x is brought very close to `0β. then it can he said that X
tends to 0. It is denoted by π₯ β 0. Here, π₯ β 0 means the value of π₯ approaches very
close to 0 but it will not be equal to 0
10) What is marginal revenue?
Ans: - The change in revenue due to small change in demand is called marginal revenue.
11) What is a base year ?
Ans: - When the price of an item is compared with the price of the same item in some
specific (fixed) year of the past, then that specific year is called the base year
12) Define correlation.
Ans: - If there are simultaneous changes in the values of two variables due to direct or
indirect cause effect then it is said that there is correlation between two variables
13) The regression co-efficient is independent of which transformation `?
Ans: - The regression co-efficient is independent from the change of origin.
14) Write the additive model of the time series
Ans: - The additive model of the time series is as follows:
π¦π‘ = ππ‘ + ππ‘ + πΆπ‘ + π π‘, where yt = variable quantity, Tt = Trend, St =
Seasonal Component, Ct = Cyclical Component, Rt = Randon Component.
15) State the number of sample points of the random experiment of tossing six balanced
coins.
Ans: - The number of sample points in the random experiment of tossing six balanced
coins isπ = 26 = 64.
16) State the relation between mean and variance of binomial distribution.
Ans: - In binomial distribution value of mean is always greater than its variance. Hence,
ππ > πππ ππ πππ < ππ
17) Standard score is independent of unit of measurement". Is this statement true or false ?
Ans: - βStandard score is independent of unit of measurement." It is a true statement.
18) Mean of a normal distribution is 13.25 and its standard deviation is 10 Estimate the
value of its third quartile?
Ans: -" In a normal distribution π3 = π + 0.675π.
Therefore, π3 = 13.25 + 0.675(10) = 13.25 + 6.75 = 20
Thus, estimate value of its third quartile is 20.
19) Find the value of limπ₯β3
π₯4β81
π₯β3
Ans: -limπ₯β3
π₯4β81
π₯β3
limπ₯β3
π₯4 β 81
π₯ β 3 [lim
π₯βπ
π₯π β ππ
π₯ β π= ππβ1]
limπ₯β3
= 4(3)4β1
limπ₯β3
= 4(3)3
= 108
20) Find π(π₯) for the function π(π₯) = 21
Ans: - Derivate of constant number is always zero. So, if function π(π₯) = 21 then
πβ²(π₯) > 0
21) What is the main purpose of index number?
Ans:- The main purpose of index number is to measure percentage relative change in
value of a variable in current period in comparison with its value in base period.
22) Define Perfect Positive Linear Correlation Coefficient.
Ans:- If the values of two random variable vary in the same direction and if one unit
increase or decrease in one variable results in the increase decrease in some fixed
proportion in the other variable then the correlation between two variables is said to be
perfect positive correlation.
23) State the linear regression model that represents the relation between two random
variables Y and X.
Ans :- Linear regression model π¦ = π + π½π₯ + π’ where, π and π½ are constants and π is
an error variable.
24) What is meant by long term component?
Ans:- The slow and steady influence on the values of a time series spread over a long
period of time is called long term component or trend.
25) If π΄ = { π₯ | 0 < π₯ < 1} πππ π΅ = {π₯ |1
4β€ π₯ β€ 3} ππππ π΄ β© π΅
Ans:-
26) State the probability distribution for Bernoulli trials.
Ans:- The probability distribution for Bernoulli trials is Binomial probability distribution
27) What is normal curve?
Ans:- The probability curve of bellβshape is called the normal curve.
28) Why is the total area of the region bounded by the normal curve and Xβ axis taken as
1?
Ans:- The normal curve is a probability curve and total probability is 1. Hence, the total area
of the region bounded by the normal curve and Xβaxis is taken as 1.
29) What is the limit of the product of a function with a constant?
Ans:- The limit of the product of a function with constant is equal to the product of the
limit of the function with the same constant. That means limπ₯βπΌ
πΎ, β« (π₯) = πΎ. πΏ where K is
the constant.
30) How the marginal cost can be obtained?
Ans:- marginal cost can be obtained by taking the derivative of the cost function with
respect to x. thus, when the product is x then margin cost = ππ
ππ₯
31) Fisher's Index Number is called an ideal index number. Write two properties in support
of the statement.
Ans:- . (I) In this Index Number, the price and quantity of base year and current year are
considered. (2) This Index Number is Geometric Mean of Laspeyre's and Paasche's index
number.
32) Write the Spearman's formula to find r.
Ans :- The Spearmanβs formula to find r is π = 1 β 6π΄π2
π(π2β1)
33) State the methods for obtaining regression lines.
Ans:- There are two methods of obtaining regression lines.
(I) Scatter diagram method.
(2) Least squares method.
34) What is forecasting?
Ans:- Forecasting is the inference based on the study of observations made on the post
phenomena for some specified point of time of the future
35) Arrange P(A u B),P(A),P(A n B),O, P(A)+(B) in the ascending order.
Ans:- Arranging in the ascending order : 0, P (A n B), P(A), P (A v B), P(A) + P(B).
36) State the conditions that p(x) becomes a probability distribution.
Ans:- If for random variable X. (I) for all the values of X of variable π π(π₯) β€ 0 and
(2) π΄π(π₯) = 1 then p (x) is the probability distribution of variable X.
37) State the relationship among the quartiles in normal distribution.
Ans:- In the normal Distribution two quartiles Q1 and Q3 are equidistant from the second
quartile Q2 = M. that means Q3 β M = M - Q1 -> Q3 + Q1=2M.
38) Mean of a normal distribution is 13.25 and its standard deviation is 10. Estimate the
value of its third quartile.
Ans:- In the normal distribution π3 = π + 0.6745π. therefore, Q3 = 13.25 +0.6745(10) =
13.25 +6.75 = 20. Thus , estimate value of its third quartile is 20.
39) If the value of the function is interminate then how can be we obtained the approximate
value of the function?
Ans:- if the value of the function is indeterminate limit can be used to approximate the
value of a function.
40) Give the meaning of increasing function.
Ans:- If h is a small positive number and if β« (π) > β« (π + β) and also β« (π) > β« (π β
β) then β« (π₯) is said to be maximum at x = a.
41) State the formula of converting chain base into number into fixed base index number.
Ans:- πΉππ₯ππ πππ π πππππ₯ ππ’ππππ =(πβπππ πππ π πππππ₯ ππ’ππππ ππ ππ’πππππ‘ π¦πππ) Γ ( πππ₯ππ πππ π πππππ₯ ππ’ππππ ππ ππππ£πππ’π π¦πππ)
100
42) State methods using in the student of correlation.
Ans:- (1) Scatter diagram method.
(2) Karl Pearsonβs product movement method.
(3) Spearmanβs rank correlation method.
43) How do you estimate dependent variable Y for the value of independent variable X?
Ans:- The estimate of demand variable Y for the value of independent variable X can be
done by regression line Y on X.
44) Give two examples of seasonal fluctuations.
Ans:- Two examples at seasonal fluctuations are
(1) Demand of woolens increases in winter.
(2) Increase in the sale of ready-made garments and fool-wears during the religious
Festivals.
45) Interpret the event A u B and A n B.
Ans:- . Event A u B means the occurrence of at least one of the events A and B. Event A n B
means the occurrence of event A and B both together.
46) By which condition parameter P of binomial distribution is negative skewness?
Ans:- In binomial distribution if π >1
2, then there is negative skewness in the distribution.
47) When it is said that "Probability distribution is normal probability distribution"?
Ans:- If the curve of the probability density function of a random variable X is bell shaped.
then the probability distribution X is called the normal distribution
48) What is the value of mean and variance in standard normal distribution?
Ans:- The standard normal distribution mean = 0 and variance = 1.
49) How is the modulus of real number?
Ans:- Modulus of any real numbers is always non-negative.
50) What is the second order derivative?
Ans:- The second order derivative of the function means the derivative of the first order
derivative of the function. It is denoted by π2π¦
ππ₯2 ππ β« "(π₯).
51) Why is Fisher's index number called an ideal index number"
Ans:- Following are the reasons for considering Fishers index number as an Ideal index
number :
(i) Geometric mean is used in its calculation.
(ii) It takes into consideration the price and quantity of both the base year and the
current year.
(iii) It satisfies the two important tests of index numbers.
(iv) It is free from bias.
52) Define 'Correlation Co-efficient'.
Ans:- The numerical measure of correlation between two variables is known as correlation
coefficient and it is denoted by r.
53) Why best In regression line method is known as least square method?
Ans:- In best lit regression line the line is selected for which the sum of square of Error's
approximation is minimum, so it is known as least square method.
54) In time series what is called long term fluctuations?
Ans:- The fluctuations which prevail on the time series for a long period of time is called
long term fluctuations or trend.
55) What is random event?
Ans:- An event depending on an unknown element (chance) is called a random event.
56) Write probability density function of binomial distribution.
Ans:- π(π = π₯) = π(π₯) =π πΆπ₯ππ₯ππβπ₯ ,
Where, π β€ π₯ β€ π, πππ π < π < 1
π₯ = ππ’ππππ ππ π π’ππππ π π ππ’π π‘πππππ .
π = ππ’ππππ ππ π π’ππππ π π ππ’π π΅πππππ’πππ π‘πππππ .
π = ππππππππππ‘π¦ ππ π π’ππππ π ππ’π π = 1 β π.
57) Draw the graph of the normal curve and locate the positions of quartiles.
Ans:- Normal Probability Curve:
58) Which value of normal variable divides the area of normal curve in two equal parts?
Ans:- . Normal curve is symmetric on both the sides for mean of normal variable X. Therefore
for π = π of normal variable divides the area of normal curve into two equal pans.
59) When it should be said that limit of a function does not exist?
Ans:- When the value of x is brought nearer to a by increasing or decreasing its value, the
value of f(x) does not approach to a particular value then it should he said that bran of a
Function does not exist.
60) What is differentiation?
Ans:- The process of obtaining derivative of a function is called differentiation.
61) Explain the main difference between explicit weight and implicit weight.
Ans:- The weight that can be expressed in number is called explicit weight and the weight
that cannot be expressed in number is called implicit weight. In implicit weighing system
more varieties of commodities are taken to which more importance are to be given.
62) Interpret r = -I.
Ans:- When r = -1 the correlation between the variables is perfect negative. Here. an
increase in the value of one variable is associated with a proportional decrease in the value
of another variable.
63) What do you mean by error of approximation?
Ans:- The difference between given value of y and estimated value of y is called error of
approximation i.e. π = π¦ β οΏ½ΜοΏ½.
64) What is random variations?
Ans:- The values in the series change due to sudden and unpredictable causes, these
changes are called as random variations.
65) Is it possible that P(A) = 0.7 and P(A U B) = 0.45 for two events A and B in a sample
space?
Ans:- For two events A and B in a sample space P(A) = 0.7 and P (A u B) 0.45; it is not
possible because it is P(A) β€ P(A u B) but here P (A u B) < P(A).
66) State the formula to find variance of discrete variable.
Ans:- Formula to find variable of discrete variable is as follows:
Variance:- π2 = π(π) = πΈ(π2) β (πΈ(π))2
Where, π΄(π2) = βπ₯π2 π(π₯π) πΆπΉ π΄(π2) = π΄π₯2π(π₯)
67) Define standard normal variate.
Ans:- If a random variable X is a normal variate with mean is and standard deviation π.
then random variable π = π₯ β π
π is called the standard normal variate.
68) For a probability distribution of standard normal variable, state the estimated limits for
the middle 50% observations.
Ans:- . For a probability distribution of standard normal variable, first quartile (Q1) = -0.675
and third quartile (Q3) = 0.675. Therefore, estimated limits for the middle 50% observation
is between - 0.675 and 0.675.
69) Define the real line?
Ans:- . The real line or real number line is a line where its points are the real numbers.
70) Find f(x) for the function f(x) = 50.
Ans:- Derivative of a constant number is always zero. So, if function f(x)=50 then f(x)=0.
71) Write the formula for obtaining the real wage?
Ans:- π πππ π€ππππ = π€πππ
πππ π‘ ππ πππ£πππ πππππ₯ ππ’ππππ Γ 100
72) State the names of different method of finding correlation.
Ans:- The different methods of finding correlation are : (1) Karl Pearsonβs product moments
methods, and (2) Spearmanβs rank correlation method.
73) What is the interval of co-efficient of determination?
Ans:- The interval of co β efficient of determination is between 0 and 1. 0 β€ R2 β€ 1.
74) Write the additive model of the time series.
Ans:- The additive model of the time series is as follows:
π¦π‘ = ππ‘ + ππ‘ + πΆπ‘ + π π‘ , π€βπππ π¦π‘ = π£πππππππ ππ’πππ‘ππ‘π¦, ππ‘ = ππππππ , ππ‘ =
π πππ ππππ πππππππππ‘, πΆπ‘ = πΆπ¦π πππππ πππππππππ‘, π π‘ = π πππππ πππππππππ‘.
75) Write definition of elementary events.
Ans:- Events consisting of only a signal elements of a sample space U are called elementary
events. Elementary events are mutually exclusive and exhaustive events.
76) What is success and failure?
Ans:- If a certain result is obtained in a random experiment it is called success and is that
certain result is not obtained, it is called failure.
77) In the standard normal distribution write the estimated values of Q1 and Q3.
Ans:- In the standard normal distribution,
π1 = π β 0675 π πππ π3 = π β 0675.
78) How is the probability curve in standard normal distribution?
Ans:- the probability curve is bell β shaped in standard normal distribution.
79) Define neighborhood of a
Ans:- Any open interval containing, π, π β π is called a neighborhood of a.
80) If y is a function x then how its differentiation can be shown?
Ans:- If y is a function of x, then is derivative is denoted by ππ¦
ππ₯.
81) What is a base year?
Ans:- An index number measure the relative change in the value of a variable as compared
to some other period. The period with which the comparison is made is known as base
year.
82) Which is the original formula for calculating correlation coefficient given by Karl
Pearson?
Ans:- π = πππ£(π₯,π¦)
ππ₯ Γ ππ¦
83) What is regression?
Ans:- Regression is the relation between two variable determined by an appropriate
mathematical function.
84) What are the components of short-term component?
Ans:- The components of short term components are:
(1) Seasonal Components (ππ‘)
(2) Cyclical Components (πΆπ‘)
(3) Irregular Components (π π‘)
85) Define conditional probability.
Ans:- suppose U is a finite sample space and A and B are two events in it. The probability of
occurrence of event B under the condition that A occurs is called the conditional probability
where,
π(π΅/π΄) = π(π΄ β© π΅)
π(π΄); π(π΄) β 0; π ππππππππ¦
π(π΄/π΅) = π(π΄ β© π΅)
π(π΅); π(π΅) β 0
86) Can mean and variance in binomial distribution be equal? Why?
Ans:- No, mean and variance in binomial distribution can not be equal. In binomial
probability distribution value of mean is always more than its variance i.e. ππ > πππ
87) State the formula to change normal random variable X into standard normal variableπ.
with parameters π and π2?
Ans:- to change normal variable X into standard normal variable Z. the formula is π = π β π
π
88) Give the values of the constants used in probability density function of normal variable.
Ans:- The value of the constant used in probability density function of normal variable is
π = 3.1416 and π = 2.7183
89) Define an interval.
Ans:- A set of real number between any two real number is an interval.
90) Findβ« "(π₯) ππ β« (π₯) = βπ₯4
.
Ans:- β« (π₯) = βπ₯4
= π₯1
4
β« β²(π₯) =1
4 π₯
1
4β1 =1
4 π₯
β3
4
β« "(π₯) = 1
4 (β
3
4) π₯
β3
4 β1
β3
16 π₯
β74 =
β3
16 π₯74
Hence, if β« (π₯) = βπ₯4
π‘βππ β« "(π₯) =β3
16 π₯74
91) The price of a commodity has increase by 5
4 times, then what is the index number of that
commodity?
Ans:- πΌππππ₯ππ’ππππ = 100 + (5
4Γ 100)
= 100 + 125 = 225
92) Give the definition of the coefficient of correlation.
Ans:- The numerical measure of correlation between two variables is known as correlation
coefficient and it is denoted by r.
93) Interpreter R2 = 1
Ans:- If R2 = 1 we can say that assumption or validity of regression is valid.
94) What does the error variable Ut show in any time series?
Ans:- The error variable Ut in a time series shows the influence of the other components
except trend.
95) What is an infinite sample space?
Ans:- A sample space with infinite number of elements means a sample space which is not
finite is called an infinite sample space.
96) What is random variable?
Ans:- Let U be a sample space of a random experiment. A function which associates a real
number with each outcomes of U, is called a random variable and is denoted by symbol X.
Symbolically, it is represented by X : U -> R.
97) What is the area of region bounded by normal curve and X β axis?
Ans:- The area of region bounded by normal curve and X β axis is 1.
98) Define normal variate.
Ans:- A random variable X, the mathematical form of its probability density function π(π₯) is
as per the normal distribution is called a normal variable.
99) If N(a;0.3) = |x -5|< b then find the value of a and b.
Ans:- Comparing π (π; 0.3) = |π₯ β 5| < 5 π€ππ‘β π(π, πΏ) = |π₯ β π| < πΏ; | π€π πππ‘ π =
5 πππ πΏ = 0.3.
100) State the rule for derivative for product of two functions of x.
Ans:- If π’ πππ π£ πππ πππππππππ‘πππππ ππ’πππ‘ππππ ππ π₯ πππ π¦ =
π’, π£, π‘βππ ππ’ππ πππ πππππ£ππ‘ππ£π πππ πππππ’ππ‘ ππ π‘π€π ππ’πππ‘πππ ππ π₯ ππ ππ ππππππ€π : ππ¦
ππ₯= π’
ππ£
ππ₯+ π£
ππ’
ππ₯
101) State the main characteristics of fixed base index number.
Ans:- In fixed base method normal year is selects as the base year and price relents of
Remaining years are found out.
102) Write the formula of Karl-Pearsonβs product moment method.
Ans:- r = Cov(x,y)
Sx Γ Sy =
π΄(x - x)Μ Μ Μ (y - y)Μ Μ Μ
βπ΄ (x - xΜ )2 Γβπ΄ (y - yΜ )2
103) When two regression lines are obtained?
Ans:- When two variables are mutually dependent two regression lines are obtained.
104) State the responsible factors of random or irregular fluctuation.
Ans:- Responsible factor of random or irregular fluctuations are as under : Political
upheaval or crises. (ii) Natural disasters like earthquake, floods, draught. (iii) Random
causes like fire, accident. strike etc.
105) Define sample space.
Ans:- A set of all possible outcomes of a random experiment is called a sample space for
that random experiment. It is denoted by the symbol U or S.
106) Who has first introduced binomial probability distribution?
Ans:- James Bernoulli introduced binomial probability distribution.
107) About which perpendicular line is the normal curve symmetric?
Ans:- The normal curve is symmetric about the perpendicular line X = π.
108) What is called the value of standard normal variable z?
Ans:- It The value of standard normal variable is called standard score or Z-score
109) Express | 3x | < 1
3 in interval form.
Ans:- Comparing |3x| <1
3 with | x - πΏ| < πΏ we get π = 0 and πΏ =
1
9.
Interval form βΆ N(π - πΏ; π + πΏ)
Putting π = 0 and πΏ = 1
9 .
|3x| <1
3 = (0-
1
9; 0 +
1
9)
= ( - 1
9;
1
9)
110) State the uses of second order differentiation.
Ans:- The second order derivative can be useful in maximization of a function. This can be
applied to be minimize cost function, maximize revenue function and maximize profits
function.
111) Give the definition of index number.
Ans:- The percentage relative change in the value of a variable quantity associated with an
item for given (current) period compared to its value in the base period is called an index
number.
112) State the different method of studying correlation.
Ans:- The different methods of studying correlation are : (i) Scatter diagram method (ii) Karl
Pearson's product moment =thud, and (iii) Spearman's rank method.
113) What is the regression line of Y on X?
Ans:- The regression line in which mathematical relation between dependent variable Y and
independent variable X is given, is called regression line of Y on X
114) State the important elements of a time series?
Ans:- There are two important elements of time series : (i) Time t and (ii) Variable Quantity
yt
115) Define independent events.
Ans:- A and B are any two events of a finite sample space βͺ. The event B is called
independent of A if the probability of occurrence of event A does not affect the probability
of occurrence of the event B i.e. if P (A/B) = P(A) and P(B/A) = P(B). then A and B are called
the mutually independent event.
116) Give illustrations of a continuous random variable.
Ans:- Illustrations of a continuous random variable are : (1) Age of a school teacher (2)
Height of a student (3) Temperature of a place (4) Weight of an employee.
117) State the value of median in standard normal distribution
Ans:- In standard normal distribution mean (is) median (M) = Mode (Ma) = 0. Therefore,
value of median is O.
118) What is the shape of normal curve?
Ans:- The shape of normal Curve is bell shaped.
119) Express 0.09 neighborhood of 0 in interval form.
Ans:- 0.09 neighborhood of 0 in interval form is express & by (0-0.09, 0+0.09) = (-0.09.0.09).
120) How will be the second order derivative of a function at x = a if function is minimum at
x = a?
Ans:- If function is minimum at x = a the second order derivative of a function at x a will be
f "(x)> O.
121) Write the name of two Methods of assigning explicit weight.
Ans:- (i) Total expenditure method
(ii) family budget method
122) State the types of correlation.
Ans:- there are three types of correlation
1) Positive correlation
2) Negative correlation
3) Lack of correlation
123) What is linear regression model?
Ans:-when the relation of linear correlation between two variables is represented by an
equation, it is called the liner regression model.
124) Give the additive model of time variable yt.
Ans:- π¦π‘ = ππ‘ + ππ‘ + πΆπ‘ + π π‘ ,
π€βπππ π¦π‘ = value of variable at time π‘.
ππ‘ = ππππππ π‘.
ππ‘ = π πππ ππππ πππππππππ‘ π‘.
πΆπ‘ = πΆπ¦π πππππ πππππππππ‘ π‘.
π π‘ = π πππππ πππππππππ‘ π‘.
125) What is a finite sample space?
Ans:- A random sample with finite number of elements is called a finite sample space.
126) State the types of random variable.
Ans:- Random variable is of two types
1) Discrete random variable and
2) Continuous random variable.
127) Which type of values of Z are left to perpendicular line at Z = 0?
Ans:- The value of Z left to perpendicular line at Z = 0 are negative.
128) State the parameters of normal distribution.
Ans:- the parameters of normal distribution are mean π and standard deviation π.
129) Express 0.05 neighborhood of 0 in interval form.
Ans:- Comparing 0.05, neighborhood of 0 with πΏ neighborhood of a, we get a = 0 and πΏ =
0.05
Interval form βΆ N(π - πΏ; π + πΏ)
Putting π = 0 and πΏ = 0.05 .
0.05 neighborhood 0 = (0 β 0.05; 0 + 0.05)
= (-0.05; 0.05)
130) How the marginal revenue can be obtained?
Ans:- marginal revenue can be obtained by taking the derivative of the revenue function
with respect to x. thus, when the product is x then margin revenue = πR
ππ₯ .