Q. 8.3 - A First Course in Probability

Question

Compute the measurement signal-to-noise ratio that is, |μ|/σ, where μ = E[X] and σ2 = Var(X) of the

following random variables:

(a) Poisson with mean λ;

(b) binomial with parameters n and p;

(c) geometric with mean 1/p;

(d) uniform over (a, b);

(e) exponential with mean 1/λ;

(f) normal with parameters μ, σ2.

Step-by-Step Solution

Answer

The values of N for each sub-parts are

$$\begin{gathered} \mathrm{a}.) \sqrt{𝜆} \\ \mathrm{b}.) \sqrt{\frac{np}{(1-p)}} \\ \mathrm{c}.) \frac{1}{\sqrt{q}} \\ \mathrm{d}.) \frac{\sqrt{\mathit{3}}\mathit{(}\mathit{a}\mathit{+}\mathit{b}\mathit{)}}{\mathit{(}\mathit{b}\mathit{-}\mathit{a}\mathit{)}} \\ \mathrm{e}.) 1 \\ \mathrm{f}.) \frac{𝜇}{𝜎} \end{gathered}$$

Given information

Let N denote signal-to-noise ratio of random variable X. 

$N=\frac{\left|𝜇\right|}{𝜎}$

where,

$E\left(X\right)=𝜇 \mathrm{and} Var\left(X\right)={𝜎}^2$

Part (a)

For Poisson distribution we have,

$N=\frac{𝜆}{\sqrt{𝜆}}=\sqrt{𝜆}$

Part (b)

For binomial distribution

$N=\frac{np}{\sqrt{np(1-p)}}=\sqrt{\frac{np}{1-p}}$

Part (c)

For geometric distribution, we have

$N=\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}}{\sqrt{{\displaystyle \raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$p^2$}\right.}}}=\frac{1}{\sqrt{q}}$

Part (d)

For uniform distribution, we have

$N=\frac{\left(a+b\right)/2}{\sqrt{{\left(b-a\right)}^2/12}}=\frac{\sqrt{3}(a+b)}{(b-a)}$

Part (e)

For exponential distribution, we have

$N=\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$𝜆$}\right.}}{\sqrt{1/{𝜆}^2}}=1$

Part (f)

For normal distibution, we have

$N=\frac{𝜇}{\sqrt{{𝜎}^2}}=\frac{𝜇}{𝜎}$