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Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2004

## Question

1. (30 pts.)

This question consists of two separate short questions relating to the structure of probability space:

(a)

Assume that $\mathcal{S}$ is the sample space of a random experiment and that $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are $\sigma$ -fields (valid event spaces) on $\mathcal{S}$ . Prove that $\mathcal{F}_{1}\cap\mathcal{F}_{2}$ is also a $\sigma$ -field on $S$ .

(b)

Consider a sample space $\mathcal{S}$ and corresponding event space $\mathcal{F}$ . Suppose that $P_{1}$ and $P_{2}$ are both balid probability measures defined on $\mathcal{F}$ . Prove that $P$ defined by $P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right),\qquad\forall A\in\mathcal{F}$ is also a valid probability measure on $\mathcal{F}$ if $\alpha_{1},\;\alpha_{2}\geq0$ and $\alpha_{1}+\alpha_{2}=1$ .

2. (10 pts.)

Identical twins come from the same egg and and hence are of the same sex. Fraternal twins have a probability $1/2$ of being of the same sex. Among twins, the probability of a fraternal set is p and of an identical set is $q=1-p$ . Given that a set of twins selected at random are of the same sex, what is the probability they are fraternal? (Simplify your answer as much as possible.) Sketch a plot of the conditional probability that the twins are fraternal given that they are of the same sex as a function of $q$ (the probability that a set of twins are identical.)

3. (30 pts.)

Let $\mathbf{X}\left(t\right)$ be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely characterized by its mean $\mu_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]$ and its autocorrelation function $R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right].$

4. (30 pts.)

Assume that $\mathbf{X}\left(t\right)$ is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function $R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\delta\left(t_{1}-t_{2}\right)$. Let $\mathbf{Y}\left(t\right)$ be a new random process defined by $\mathbf{Y}\left(t\right)=\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds$, where $T>0$ .

(a)

What is the mean of $\mathbf{Y}\left(t\right)$ ?

(b)

What is the autocorrelation function of $\mathbf{Y}\left(t\right)$ ?

(c)

Write an expression for the second-order pdf $f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right)$ of $\mathbf{Y}\left(t\right)$ .

(d)

Under what conditions on $t_{1}$ and $t_{2}$ will $\mathbf{Y}\left(t_{1}\right)$ and $\mathbf{Y}\left(t_{2}\right)$ be statistically independent?