Covariance vs Correlation: Difference, Formula, Examples and Comparison Introduction Covariance and correlation are two important statistical concepts used to measure the relationship between two variables. They help answer questions such as: Does advertising increase sales? Does study time affect exam marks? Are height and weight related? Although both measure relationships between variables, covariance and correlation have important differences in interpretation and usage. What is Covariance? Covariance measures the direction in which two variables change together. $$ Cov(X,Y)= \frac{\sum(X_i-\bar X)(Y_i-\bar Y)} {n} $$ Interpretation of Covariance Positive Covariance When: $$ Cov(X,Y)>0 $$ Both variables increase or decrease together. Negative Covariance $$ Cov(X,Y) One variable increases while the other decreases. Zero Covariance $$ Cov(X,Y)=0 $$ There is no linear relationship between v...

Question Find the integrating factor of the differential equation $$ \frac{dy}{dx} + \frac{x}{1-x^2}y = x\sqrt{y} $$ Solution The given differential equation is $$ \frac{dy}{dx} + \frac{x}{1-x^2}y = x\sqrt{y} $$ Since the equation contains the term $\sqrt{y}$, it is not linear. Let $$ v=\sqrt{y} $$ Then, $$ y=v^2 $$ Differentiating, $$ \frac{dy}{dx} = 2v\frac{dv}{dx} $$ Substitute these values into the given equation: $$ 2v\frac{dv}{dx} + \frac{x}{1-x^2}v^2 = xv $$ Divide both sides by $2v$: $$ \frac{dv}{dx} + \frac{x}{2(1-x^2)}v = \frac{x}{2} $$ This is now a linear differential equation of the form $$ \frac{dv}{dx}+P(x)v=Q(x) $$ where $$ P(x)= \frac{x}{2(1-x^2)}. $$ The integrating factor is $$ \text{I.F.} = e^{\int P(x)\,dx} = e^{\int \frac{x}{2(1-x^2)}dx} $$ Let $$ u=1-x^2 $$ Then, $$ du=-2x\,dx $$ Therefore, $$ \int \frac{x}{2(1-x^2)}dx = -\frac14 \int \frac{du}{u} $$ Hence, $$ = -\frac14\ln|1-x^2| $$ ...

  Line Integral Solution Solution $ \mathbf{F}(x,y)=-x\hat{i}+y\hat{j} $ $$ \int_C \mathbf{F}\cdot d\mathbf{r} $$ $$ =\int_C(-x\,dx+y\,dy) $$ Parameterization: $$ x=\cos t,\qquad y=\sin t $$ $$ dx=-\sin t\,dt,\qquad dy=\cos t\,dt $$ $$ \int_{0}^{\pi/4} \left[ (-\cos t)(-\sin t) + (\sin t)(\cos t) \right]dt $$ $$ = \int_{0}^{\pi/4} 2\sin t\cos t\,dt $$ where $$ 0\le t\le\frac{\pi}{4} $$ $$ 2\sin t\cos t=\sin2t $$ $$ =\int_{0}^{\pi/4}\sin2t\,dt $$ $$ =\left[-\frac12\cos2t\right]_{0}^{\pi/4} $$ $$ =-\frac12\cos\frac{\pi}{2} +\frac12\cos0 $$ $$ =0+\frac12 =\frac12 $$ $$ \boxed{\int_C \mathbf{F}\cdot d\mathbf{r}=\frac12} $$ Option A GATE EC Previous Year Papers with Solutions → Electronics and Communication Study Material →

Download Question Paper                   2025 | 2024 | 2023 | 2022 | 2021 | 2020 GATE - EC 2025 Answers with Explanations  Q.1 Current population 11,02,500 increasing 5% per annum Let assume 2 years age, the population were, x then, x*(1+5/100)^2 = 11,02,500 Or, x = 1000000 Option C Q.2 p/q + q/p = 3 Or, (p/q)^2 + (q/p)^2 = (p/q + q/p)^2 - 2 Or, (p/q)^2 + (q/p)^2 = 9-2=7 Option B Q.3 Option C Q.4 Option C Q.5 Option B Q.6 Option C Q.7 Option D Q.8 Total Volume = 1/3*pi*r^2*l = 1/3*pi*1*1 = pi/3 Option A Q.9 Option C Q.10 The area of the triangle is  1/2*3x*3x = 9/2*x^2 The area of the regular convex hexagon is  9/2*x^2 - 3*1/2*x*x = 6/2*x^2 Or, area of hexagon: triangle = 2:3 Option D Electronics and Communication (EC) Q.1 $ \mathbf{F}(x,y)=-x\hat{i}+y\hat{j} $ $$ \int_C \mathbf{F}\cdot d\mathbf{r} $$ $$ =\int_C(-x\,dx+y\,dy) $$ Parameterization: $$ x=\cos t,\qquad y=\sin t $$ $$ dx=-\sin t\,dt,\qqua...

  Understanding OFDM Overhead: Why Real-World 5G Speeds Differ from Theory Why Internet Isn't as Fast as the Theory: Understanding Overhead Have you ever wondered why a 100 Mbps wireless connection often delivers only 70 or 80 Mbps in a speed test? In the world of OFDM (Orthogonal Frequency Division Multiplexing) —the technology powering 5G, LTE, and Wi-Fi 6—the difference between theoretical limits and real-world performance is caused by Overhead . What is Overhead in OFDM? In telecommunications, overhead is any data transmitted that isn’t the actual user payload. It is the "management tax" required to keep a wireless connection stable, synchronized, and error-free. The 4 Main Types of Practical Overhead 1. Frequency Guard Bands Wireless standards cannot use 100% of their allocated frequency. In a 20 MHz channel, roughly 10% is left empty at the edges. This acts as a buffer to prevent ...

  1. Noise Remover (Dead-Zone Clipper - Image 1) Vi Vo V1 V2 Signal Amp (Vp): Noise Level: Deadzone V1: Deadzone V2: --- Noisy Input    — Cleaned Output (Series) 2. Dual Parallel Clipper (Shunt - Image 2) Vi Vo ...

v_i V_R v_o Peak Amplitude (V): Ref Voltage V R (V): Diode Barrier V d (V): --- Input v i    — Output v o    - - Threshold (V R + V d ) Practical Tip: If V R is negative, the battery symbol in reality would be flipped. v_o Peak Input (V p ): Ref Battery (V R ): Barrier (V d ): ...

A symbol stream contains alternate QPSK and 16-QAM symbols. If symbols from this stream are transmitted at the rate of 1 mega-symbols per second, the raw (uncoded) data rate is _______ mega-bits per second (rounded off to one decimal place).   Answer:  Since alternative QPSK and 16-QAM used, data rate of uncoded data is (2+4)/2 = 3 Mbps Number of bits in one QPSK symbol is 2 and number of bits in one 16-QAM symbol is 4 Read more about Modulation Techniques → GATE EC Previous Year Papers with Solutions → Electronics and Communication Study Material →