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Important Formula for RBI Grade B DEPR Phase - 1 - Statistics and Econometrics

Karl Pearson's Correlation Coefficient

For Population:

\[{\rho _{XY}} = \frac{{cov(X,Y)}}{{{\sigma _X} \cdot {\sigma _Y}}}{\rm{ }}\]

For Sample:

\[\mathop r\nolimits_{XY} = \frac{{{s_{YX}}}}{{{s_X} \cdot {s_Y}}}\] \[\left\{ \begin{array}{l}Where,\\{s_{YX}} = \frac{{\sum {\left( {X - \overline X } \right)\left( {X - \overline X } \right)} }}{n}\\{s_X} = {\rm{ }}\sqrt {\frac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}} \\{s_Y} = \sqrt {\frac{{\sum {{{\left( {Y - \overline Y } \right)}^2}} }}{n}} \end{array} \right.\]

Substituting the formula of Covariance and Standard Deviation: \[\mathop r\nolimits_{XY} = \frac{{\frac{{\sum {\left( {X - \overline X } \right)\left( {X - \overline X } \right)} }}{n}}}{{\sqrt {\frac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}} \cdot \sqrt {\frac{{\sum {{{\left( {Y - \overline Y } \right)}^2}} }}{n}} }}\] Cancelling n: \[\mathop r\nolimits_{XY} = \frac{{\sum {\left( {{X_i} - \overline X } \right)\left( {{Y_i} - \overline Y } \right)} }}{{\sqrt {\sum {{{\left( {{X_i} - \overline X } \right)}^2} \cdot \sum {{{\left( {{Y_i} - \overline Y } \right)}^2}} } } }}\] Deviation Form: \[\mathop r\nolimits_{XY} = \frac{{\sum {\mathop x\nolimits_i \mathop y\nolimits_i } }}{{\sqrt {\sum {\mathop x\nolimits_i^2 } } \cdot \sqrt {\sum {\mathop y\nolimits_i^2 } } }}\] Formula without Mean and Deviation: Formula without Mean and Deviation: \[\mathop r\nolimits_{XY} = \frac{{n\sum {\left( {{X_i}{Y_i}} \right) - \left( {\sum {{X_i}} } \right)\left( {\sum {{Y_i}} } \right)} }}{{\sqrt {n\sum {X_i^2 - {{\left( {\sum {{X_i}} } \right)}^2}} } \cdot \sqrt {n\sum {Y_i^2 - {{\left( {\sum {{Y_i}} } \right)}^2}} } }}\]

Spearman's Rank Correlation Coefficient:

\[r' = 1 - \frac{{6\sum {{D^2}} }}{{n\left( {{n^2} - 1} \right)}}\] Where,
D = Difference between the ranks of corresponding pairs of X and Y

Simple Linear Regession Model:

For Population

\[\begin{array}{l}{Y_i} = {b_0} + {b_1}{X_i}\\{Y_i} = {b_0} + {b_1}{X_i} + u\end{array}\]

For Sample

\[{{\hat Y}_i} = {{\hat b}_0} + {{\hat b}_1}{X_i}\]

OLS Estimation

\[\begin{array}{l}{Y_i} - {{\hat Y}_i} = {e_i}\\Minimize \to {e_i} = {Y_i} - {{\hat Y}_i}\\Min\sum {\mathop e\nolimits_i^2 } = \sum {\mathop {\left( {{Y_i} - {{\hat Y}_i}} \right)}\nolimits^2 } \\Min\sum {\mathop e\nolimits_i^2 } = \sum {\mathop {\left( {{Y_i} - {{\hat b}_0} + {{\hat b}_1}{X_i}} \right)}\nolimits^2 } \\Slope:\\\mathop {\hat b}\nolimits_1 = \frac{{n\sum {\mathop X\nolimits_i \mathop Y\nolimits_i - \sum {\mathop X\nolimits_i \sum {\mathop Y\nolimits_i } } } }}{{n\sum {\mathop X\nolimits_i^2 - \mathop {\left( {\sum {\mathop X\nolimits_i } } \right)}\nolimits^2 } }}\\\mathop {\hat b}\nolimits_1 = \frac{{\sum {\mathop x\nolimits_i \mathop y\nolimits_i } }}{{\sum {\mathop x\nolimits_i^2 } }} = \frac{{{\mathop{\rm cov}} (X,Y)}}{{\mathop \sigma \nolimits_X^2 }}\\Intercept:\\\mathop {\hat b}\nolimits_0 = \overline Y - {{\hat b}_1}\overline X \end{array}\]

Hypothesis Testing of estimators:

Null Hypothesis = Estimators are not statistically more than zero.
\[\begin{array}{l}\mathop t\nolimits_0 = \frac{{{{\hat b}_0} - {b_0}}}{{{s_{{{\widehat b}_0}}}}}\\\mathop t\nolimits_1 = \frac{{{{\hat b}_1} - {b_1}}}{{{s_{{{\widehat b}_1}}}}}\end{array}\]

Variance of estimators when population variance is given:

\[\begin{array}{l}Var{\rm{ }}{\widehat b_0} = \sigma _u^2\frac{{\sum {X_i^2} }}{{n\sum {x_i^2} }}\\Var{\rm{ }}{\widehat b_1} = \sigma _u^2\frac{1}{{\sum {x_i^2} }}\end{array}\]

Estimator of Population Variance (\(\sigma _u^2\)):

\[{s^2} = \hat \sigma _u^2 = \frac{{\sum {e_i^2} }}{{n - k}}\]

Variance of estimators when population variance is not given:

\[\begin{array}{l}s_{{{\widehat b}_0}}^2 = \frac{{\sum {e_i^2} }}{{n - k}} \cdot \frac{{\sum {X_i^2} }}{{n\sum {x_i^2} }}\\s_{{{\widehat b}_1}}^2 = \frac{{\sum {e_i^2} }}{{n - k}} \cdot \frac{1}{{\sum {x_i^2} }}\end{array}\]

Coefficient of Determination:

\[\mathop R\nolimits^2 = \frac{{\sum {\hat y_i^2} }}{{\sum {y_i^2} }} = 1 - \frac{{\sum {e_i^2} }}{{\sum {y_i^2} }} = \mathop {\hat b}\nolimits_1 \frac{{\sum {{x_i}{y_i}} }}{{\sum {y_i^2} }}\]

Relationship between Correlation Coefficient & Coefficient of Determination:

\[r = \sqrt {{R^2}} \]

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Freelancing |Benefits| Opportunities|

How can a Freelancer cut down your cost?

India’s gig economy is growing at an exponential rate and it is projected to grow to around 500 USD in next three years. According to a report by Deloitte, over 50% organizations expect an increase in the number of Freelancers. This implies that India’s appetite for Gig economy is getting bigger and bigger and freelancers are an integral part of it. Companies are now laying stress on hiring an expert for a specific period of time which opens the gate for a Freelancer. A research by PayPal indicates that India’s contribution to global freelance workforce is concrete and it has been observed that out of every 100 freelancers, 25 are Indians covering all the diversified sectors such as content writing, academic writing, blog writing, Information & technology, web designing, graphic designing, animation etc. Amid the ongoing pandemic situation generated by COVID -19 virus, more and more employers are stressing on working with Freelancers, which paves the way for increasing digitization.

A Freelancer value preposition can be seen in terms of the specialization, time consumed in completing a task, value charged against the work etc. Below are some of the key advantages of a freelancer which helps an organization in bring down the cost:

  • Cost effectiveness: Cost reduction is first and the foremost advantage that a freelancer brings along with itself. Hiring an employee traditionally, will lead to paying a fixed monthly salary, medical benefits, provident fund, gratuity etc. irrespective of the fact that whether the work is done or not. There have been instances where an employee gets complacent and did not perform as the expectations after the salary negotiation. However, this is usually not the case with a Freelancer as there payment is primarily dependent on deliverables and performance.
  • Higher RoI (Return on Investment): When organizations hire a freelancer, they end up enjoying higher returns, thus leaving behind remunerations and cherishing optimization. This virtue of a higher RoI is attributed to the fact that a cost of hiring a freelancer is way to lower than hiring a full- time employee. There is no commission involved in hiring a freelancer. A freelancer is hired in a very short span of time as compared to a full time employee as the there is a standard lengthy procedure involved in latter’s hiring.  
  • Employment Handling: A greater degree of ease has been witnessed by the employer be it a start-up or an organization in terms of employment handling in case of a freelancer. Organizations have the flexibility of paying a freelancer only for a specific period of time, largely dependent on the project requirements. Another benefit of hiring a freelancer is that the organization doesn’t have to worry about the downtime because in the event of decline in business, the work shall not be send to them. The process of termination of service looks less cumbersome when compared to the termination process of a full time employee which is resource intensive. In the event of replacement, a freelancer can be replaced at much ease than compared to a full time employee.
  • Quality: A freelancer has a huge array of experience and exposure. They are far more acclimatized to industry norms than their counter parts within the organization. They have the experience of dealing with multiple clients, their requirements which make them more aware of the market condition and requirements.  They are open minded about bad times, hence they can perform better in a pressure scenario which makes them eligible to be a part of organization’s growing talent pool. For a freelancer, nurturing relationship is of foremost importance and this also an integral part of any organization’s business policy. A full time
  • Global Reach: A freelancer brings along with him a diversified plethora of experience. Opting for a freelancer enables an organization to expand it’s reach in terms of location. This offers an organization to go across boundaries and hire the best from the global pool of talent for a specific period of time to complete the task. Thus, hiring a freelancer reduces the cost of training which ultimately eliminates the risk which comes in with hiring an employee who needs to be trained for a specific project and turns into a liability once the project is completed.   

Below are some of the top freelancing websites in India which provide a wide range of opportunities to nation’s increasing freelance work force:

  1.  Rozgaar India: Rozgaar India is India’s first and very own online market place which brings the employer and the employee together. Launched under the ambit of Skill India & digital India mission, with an objective of increasing digitization by providing work opportunity online to every freelancer in India coming from any field of work, this platform is designed to cater to the needs of Indian employers as well as the employee. This platform has a wide variety of competent freelancers with great skill sets and field experience who can cater to the industry requirements of any field including academics, IT, animation, engineering, coding, singing etc. The amplitude of quality projects available on Rozgaar India is very high which makes it every freelancer’s first choice.  The platform continuously refines and updates itself to keep itself align with the dynamics of the game. With its mobile application compatible for IOs and Android mobile phones, Rozgaar India is readily available 24*7 to every Indian. Hence, Rozgaar India is the most sought after, trust worthy source in India for freelancing part time and full time jobs which has an inclusive approach for both employer and employee.
  2. Fiverr: It is online Israeli online market place for freelance services. It has been ranked among the top most popular sites worldwide. Having the reputation of providing work from almost every niche of the market at minimum cost of 5$. Fiverr also provides rating after successful completion of work which gives a fair idea about the freelancer to future employers about the quality of work to be expected.  
  3. Upwork: It is one of the most popular freelancing sites in India. One has to be a little patient while going live on Upwork , promoting the work will take time here but once done, you can surely earn a living here. Trusted with more than 5 million users, Upwork provides work opportunity to everyone irrespective of their area of specialization.
  4. Freelancers: It has the reputation of being the oldest and a little hard to navigate through but the quality and authenticity of projects available here makes Freelancer, one of the most credible online platform for Freelancers.
  5. Truelancer: This website is known for its ‘get a job in no time’ approach, it has the reputation of getting a job within one week from any field including Software development, design, data entry, finance etc.


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