Current Affairs 9 January 2019 - करंट अफेयर्स 9 जनवरी 2019 - Daily Current Affairs - Current Affairs Hindi

Q.1- केंद्र सरकार ने आर्थिक रूप से पिछड़े सवर्णों को कितने परसेंट आरक्षण देने की घोषणा की है-

Ans- 10%

*केंद्र सरकार ने 7 जनवरी 2019 को एक बहुत ही महत्वपूर्ण निर्णय लिया है. इसके अनुसार आर्थिक रूप से पिछड़े सवर्णों यानी सामान्य वर्ग के लिए सरकारी नौकरियों और शिक्षण संस्थानों में 10% आरक्षण दिया जाएगा.
*यह आरक्षण सामान्य श्रेणी के उन लोगों को दिया जाएगा, जिनकी सालाना आय 8 लाख रुपये से कम है.
*इस विधेयक के लिए संविधान की धारा 15 और 16 में संशोधन किया जाएगा.
*केंद्र सरकार जल्दी ही इसके लिए लोकसभा में संशोधन विधेयक पेश करेगी.

Q.2- भारत ने किस देश में चाबहार बंदरगाह का परिचालन शुरू कर दिया है?

Ans- ईरान

*भारत सरकार ने ईरान में शाहिद बेहेश्ती बंदरगाह (चाबहार) के एक हिस्से का परिचालन अपने हाथ में ले लिया है.
*चाबहार बंदरगाह ईरान के सिस्तान बलूचिस्तान प्रांत में स्थित एक बंदरगाह है, जो कि रणनीतिक दृष्टि से बहुत ही महत्वपूर्ण है.
*चाबहार बंदरगाह से भारत के कारोबारियों को बहुत लाभ होगा. अब भारतीय कारोबारी अपना सामान सीधे ईरान तक बिना किसी व्यवधान के भेज पाएंगे.
*चाबहार बंदरगाह के शुरू होने पर भारत का निर्यात का समय घट जाएगा तथा निर्यात पर होने वाला खर्चा भी लगभग एक तिहाई तक कम हो जाएगा.

Q.3- कौन सा खिलाड़ी होपमैन कप जीतने वाला पहला खिलाड़ी बना है?

Ans- रोजर फेडरर

*स्विट्जरलैंड के रोजर फेडरर ने जर्मनी को 2 -1 से हराकर होपमैन कप का खिताब जीत लिया. 
*इसके साथ ही रोजर फेडरर 3 होपमैन कप जीतने वाले पहले खिलाड़ी बन गए हैं.

Q.4- सुप्रीम कोर्ट ने किस सीबीआई डायरेक्टर को पद से हटाने का केंद्र का फैसला निरस्त करके बहाल किया है?

Ans- आलोक वर्मा

*सीबीआई डायरेक्टर आलोक वर्मा को पद से हटाकर छुट्टी पर भेजने का केंद्र सरकार का फैसला सुप्रीम कोर्ट ने निरस्त कर दिया है.
*लेकिन जब तक उनके खिलाफ भ्रष्टाचार के आरोपों की जांच पूरी नहीं हो जाती तब तक उन्हें नीतिगत फैसले लेने से रोक दिया गया है.

Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

What is Marginal Probability Density Function (Marginal PDF) or Marginal Densities?

When the PDFs fx(x) and fy(y) for any single random variable are obtained from the joint PDF, in that case fx(x) and fy(y) are called as marginal PDF or marginal densities.

Watch the Complete Video Here

 

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Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

 Proof-

Now to find the CDF of random variable X, the value of other random variable Y, does not matter.

Similarly we can get-

Here fx(x) and fy(y) are known as marginal PDF or simply marginal densities as both of these PDFs are obtained from the joint PDF.

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Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

What is Conditional Probability Density Function (Conditional PDF)?

A probability density function is known as conditional PDF, when one random variable out of two random variables has a fixed value. Here suppose we have two random variables X and Y, and X has a fixed value equal to x. 
In this case in the conditional PDF of Y when X=x is given as-

In the same way, Conditional PDF of X when random variable Y takes a fixed value Y=y is given as-

Watch the Complete Video Here

Properties of Conditional Probability Density Function (Conditional PDF)

Property 1- Conditional PDF is a non-negative function.

As conditional PDF is a ratio of two PDFs and we know that PDF is a non-negative function. Therefore the ratio of two non-negative PDFs is also non- negative function.

Property 2- Area under the conditional PDF is unity i.e. one.
This property mathematically can be expressed as-

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Property 3- Conditional probability density function (conditional PDF) reduces to marginal density if random variables X and Y are statistically independent.

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Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Joint Cumulative Distribution Function is also known as Joint Distribution Function or Combined CDF
Here we will discuss the CDF for two random variables X and Y.

Watch the Complete Video Here

Definition of Joint Distribution Function (Combined CDF)

Joint CDF : [FXY(x,y)], of two random variables X and Y is defined as the probability that the random variable 'X' is less than or equal to a specified value 'x' and the random variable 'Y' is less than or equal to a specified value 'y'.

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Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

This definition of joint commutative distribution function (Joint Distribution Function) can be represented mathematically as-

Properties of Joint Cumulative Distribution Function (Combined CDF)

Property 1- The joint cumulative distribution function is a monotone non-decreasing function of both x and y.

Property 2- Combined CDF is a non-negative function.
=>   Fxy(x,y) ≥ 0
It is defined as the probability in the joint sample space of random variables and probability lies between 0 and 1.
Therefore joint cumulative distribution function (Joint CDF) also lies between 0 and 1.
=> It is a non-negative function

Property 3- Joint cumulative distribution function is always continuous everywhere in the xy-plane.

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Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

What is Joint Probability Density Function or Joint PDF?

Joint PDF is simply the PDF of two or more random variables.
The joint probability density function of any two random variables X and Y can be defined as the partial derivative of the joint cumulative distribution function, with respect to dummy variables x and y.
Mathematically-

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Joint PDF Formula

Joint PDF Formula

Now we will discuss the properties of joint probability density function (joint PDF)

Properties of Joint Probability Density Function (Joint PDF)

Property 1- Joint PDF is non-negative.

Joint PDF Property

Since joint PDF is a derivative of joint cumulative distribution function (Joint CDF), which is also a non negative function.
Therefore joint PDF is always positive.

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Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

Property 2- The joint PDF is continuous everywhere as the joint CDF is continuous and we know that it is the derivative of joint CDF.

Property 3- The total volume under the surface of joint PDF is equal to Unity.

Joint PDF Property

Now we will discuss an important property of two statistically independent random variables X and Y.

Statistically Independent Random Variables X and Y

For two statistically independent random variables X and Y-

Statistically independent random variables X and Y

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Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Proof-

Statistically Independent Random Variables X and Y

Relation between Probability and Joint PDF

Relation between Probability and Joint PDF

On extending this relation to two random variables X and Y-

Relation between Probability and Joint PDF for two Random Variables

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Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

Relationship between joint PDF and Probability for statistically independent random variables X and Y

If two random variables X and Y are statistically independent, then the joint PDF of X and Y is given as the product of two separate PDFs.

Relationship between joint PDF and Probability for statistically independent random variables X and Y

Relationship between joint PDF and Probability for statistically independent random variables 
X and Y

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Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

Let's first understand, what is Cumulative Distribution Function (CDF) and it's Definition-

What is Cumulative Distribution Function (CDF)?

Definition of CDF-

The Cumulative Distribution Function (CDF) of a random variable 'X' may be defined as the probability that the random variable 'X' takes a value 'Less than or equal to x'.
Mathematically it can be represented as-

CDF Formula

Cumulative Distribution Function (CDF) may be defined for-
#Continuous random variables and
#Discrete random variables

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Cumulative Distribution Function (CDF) other Names

#Probability distribution function of the random variable
#Distribution function of the random variable
#Cumulative probability distribution function

Properties of Cumulative Distribution Function (CDF)

Following image discusses the 3 Properties of Cumulative Distribution Function (CDF)

Properties of CDF (Cumulative Distribution Function Properties)

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Cumulative Distribution Function (CDF) for discrete random variables

If 'X' is a discrete random variable, then it takes on values at discrete points.
Therefore CDF can be defined for this case as shown in the image below-

CDF for Discrete Random Variable Explanation 

So the CDF for a discrete random variable for the complete range of x can be defined as shown in the image below-

CDF for Discrete Random Variable

=> Cumulative Distribution Function (CDF) of a discrete variable at any certain event is equal to the summation of the probabilities of random variable upto that certain event. 
As x varies from -∞ to ∞ the graph of CDF i.e. Fx(x) resembles a staircase with upward steps having height P(X=xj) at each x=xj.
But note one thing that the graph of Fx(x) CDF remains constant between the two steps or events.

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Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

What is Probability Density Function (PDF)?

The derivative of Cumulative Distribution Function (CDF) w.r.t. some dummy variable is called as probability density function (PDF).
Probability density function can be defined mathematically as-

Relation between PDF and CDF (Formula of PDF)

Now we will discuss the properties of probability density function. The derivation of properties of PDF is also provided here.

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Properties of Probability Density Function (PDF) with Derivation

Property 1- Probability density function is always non-zero for all values of x.

PDF Property 1 with Proof

Property 2- The area under the PDF curve is always equal to Unity i.e. one.

PDF Property 2 With Proof

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Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Property 3- It is possible to get Cumulative Distribution Function (CDF) by integrating PDF.

PDF Property 3 With Proof

Property 4- Probability of the event {x1< X<= x2} is given by the area under the PDF curve in 
{x1< X<= x2} range.

PDF Property 4 With Proof

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Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof