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I have a problem as follows: $Y_k=N_k+AS_K \quad ,k=1,\ldots,n.$$$where $\underline{N} \sim N(\underline{0},I)$ and where $S_1,\ldots,S_n$ are i.i.d. random variables, independent of $\underline{N}$ and each taking on the values $+1$ and $-1$ with equal probabilities of $\frac{1}{2}$. I want to obtain pdf of $\underline{Y}$ ? $A$ is constant value. Can anyone help me! Thank you.

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    Thanks alot for your answers!2012-11-25

2 Answers 2

1

The entries $Y_k$ of $\underline{Y}$ are i.i.d. If the density of each $N_k$ is $g$, the density of each $Y_k$ is the function $f$ defined by $f(x)=\frac12(g(x+A)+g(x-A))$.

2

You can directly use the independence condition. $ P( Y_k \leq x ) = P( N_k + AS_k \leq x ) \\ = P( N_k + AS_k \leq x | S_k = -1 )P( S_k = -1 ) + P( N_k + AS_k \leq x | S_k = 1 )P( S_k = 1 ) = P( N_k - A \leq x | S_k = -1 )P( S_k = -1 ) + P( N_k + A \leq x | S_k = 1 )P( S_k = 1 ) = P( N_k - A \leq x )P( S_k = -1 ) + P( N_k + A \leq x )P( S_k = 1 ) \\ = P( N_k \leq x + A )P( S_k = -1 ) + P( N_k \leq x - A )P( S_k = 1 ) \\ = \frac{1}{2}( \int_{-\infty}^{x+A} f(t)dt + \int_{-\infty}^{x-A} f(t)dt \: ) \\ = \frac{1}{2}( \int_{-\infty}^{x} f(t - A)dt + \int_{-\infty}^{x} f(t+A)dt \: ) \\ = \int_{-\infty}^{x} \frac{f(t - A) + f(t + A)}{2} dt $

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    I just didn't notice that, thanks $f$or the remark2012-11-26