Expectation and Even PDF

By Definition of Expectation of Random Variable:

$$ E(X)=\int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$

If $f_X(x)$ is Even, Then $E(X)=0$, Provided The Integral Exists, An Exception Being $\textbf{Cauchy Distribution}$ , Whose Mean Doesn't Exist.

Now the Question is, Will the PDF of Random Variable is Even ,When $E(X)=0$, I am Unable to Solve/Prove This. I Encourage all of you to Solve This...

Conditional Distribution

Consider two random Variables $X$ and $Y$ whose Joint PDF is:
\begin{align*}
 f_{XY}(x,y) &=2(x+y),\; 0 \leq y \leq  1,\; 0 \leq x \leq y \\
&=0,\; Else
\end{align*}

Find $$f_{Y|X}(y|x) $$

Independence and UnCorrelated

We Know that if Two Random variables $X$ and $Y$ are  Independent Statistically, Then

$$ E(XY)=E(X)E(Y)$$, i.e., They are UnCorrelated. It Doesnt Mean that Dependent Random Variables are Always Correlated. Lets Think of Some Examples with Dependent Random Variables being UnCorrelated.

Coin Tossing and Mutual Information

There are Two Coins viz., A fair Coin and a Two Headed Coin. A Coin is Selected at Random for which the Random Variable is Denoted as X, and Selected Coin is Tossed Twice, and Number of Heads is Recorded which is Denoted as Random variable Y.

Find $$ I(X;Y)$$

Log Normal Distribution

if $X$ is Gaussian Distributed Random variable i.e., if
\begin{align*}
X \sim  \mathcal{N}(\mu,\sigma^2) \\
Y=e^X\\
\end{align*}
Y is said to be Log Normally Distributed i.e.,
\begin{align*}
Y \sim Log-\mathcal{N}(\mu,\sigma^2)
\end{align*}
 If MGF of X is Given i.e., $\Phi_X(s)$ is Given.

Find the nth Moment of Y, where $n \in \mathbb{Z}$, Without Finding the PDF of Y

A Very Interesting Probability Problem

Consider a Line Segment of length $L$. Randomly Select a Point on Each Side of the Midpoint of the Line Segment. What is The Probability that The Distance Between the Two Points Randomly Selected is Greater Than $\frac{L}{3}$

Testing Math

$$\sum_{k=1}^{N} k= \frac{N(N+1)}{2}$$

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