Independent distributions I have a confusion regarding the notion of independence of distributions.
what is meant by saying two distributions are independent..? suppose I have two normal distributions with means 1,-1 and variances 1/2, 1/2 respectively. then is the sum of the pdf is also a pdf of normal distribution or not..? 
 A: Let $X\sim\mathcal{N}(1,\frac{1}{2})$ and $Y\sim\mathcal{N}(-1,\frac{1}{2})$ be two normal distributed random variables with mean $1$ and $-1$ respectively, both having variance $\frac{1}{2}$. We haven't said anything about the relationship between the two variables yet, and unless we do so, we can not say anything about $X+Y$ for sure. Let us therefore introduce independency:
We say that $X$ and $Y$ are independent if
$$
P(X\in A,X\in B)=P(X\in A)P(X\in B),\quad \text{for all }\,A,B\in\mathcal{B}(\mathbb{R}),
$$
which is equivalent to saying that 
$$
P(X\leq x,Y\leq y)=P(X\leq x)P(Y\leq y),\quad x,y\in\mathbb{R},
$$
i.e. the joint distribution function equals the product of the two marginal distribution functions. If $X$ and $Y$ are independent, then $X+Y$ is again normal distributed (this is a special property of the normal distribution) with mean 
$$
E[X+Y]=E[X]+E[Y]=1+(-1)=0,
$$
and variance
$$
\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)=1.
$$


then is the sum of the pdf is also a pdf of normal distribution or not..?

The pdf of the sum of two variables is not the sum of the individual pdfs. The sum of two pdf's need not even be a pdf.
