7.8 QE 2004 January
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 $ .
Answer
• Because $ P_{1} $ and $ P_{2} $ are valid probability measures, we know that they satisfy the axioms of probability:
1. $ P_{1}\left(A\right)\geq0 $ and $ P_{2}\left(A\right)\geq0 $ , $ \forall A\in\mathcal{F}\left(\mathcal{S}\right) $ .
2. $ P_{1}\left(\mathcal{S}\right)=1 $ and $ P_{2}\left(\mathcal{S}\right)=1 $ .
3. If $ A_{1} $ and $ A_{2}\in\mathcal{F}\left(\mathcal{S}\right) $ are disjoint events, then $ P_{1}\left(A_{1}\cup A_{2}\right)=P_{1}\left(A_{1}\right)+P_{1}\left(A_{2}\right) $ and $ P_{2}\left(A_{1}\cup A_{2}\right)=P_{2}\left(A_{1}\right)+P_{2}\left(A_{2}\right) $ .
4. If $ A_{1},A_{2},\cdots,A_{n},\cdots\in\mathcal{F}\left(\mathcal{S}\right) $ is countable collection of disjoint events, then $ P_{1}\left(\cup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P_{1}\left(A_{i}\right) $ and $ P_{2}\left(\cup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P_{2}\left(A_{i}\right) $ .
• Now, we check each condition to become a valid probability measure:
1. $ P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right)\geq0 , \forall A\in\mathcal{F}\left(\mathcal{S}\right) $ .
– $ \because\alpha_{1}\geq0,\;\alpha_{2}\geq0,\; P_{1}\left(A\right)\geq0,\text{ and }P_{2}\left(A\right)\geq0 $ .
2. $ P\left(S\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right)=\alpha_{1}+\alpha_{2}=1 $ .
3. If $ A_{1} $ and $ A_{2}\in\mathcal{F}\left(\mathcal{S}\right) $ are disjoint events, then $ P\left(A_{1}\cup A_{2}\right)=\alpha_{1}P_{1}\left(A_{1}\cup A_{2}\right)+\alpha_{2}P_{2}\left(A_{1}\cup A_{2}\right)=\alpha_{1}\left\{ P_{1}\left(A_{1}\right)+P_{1}\left(A_{2}\right)\right\} +\alpha_{2}\left\{ P_{2}\left(A_{1}\right)+P_{2}\left(A_{2}\right)\right\} $$ =\alpha_{1}P_{1}\left(A_{1}\right)+\alpha_{2}P_{2}\left(A_{1}\right)+\alpha_{1}P_{1}\left(A_{2}\right)+\alpha_{2}P_{2}\left(A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right). $
4. If $ A_{1},A_{2},\cdots,A_{n},\cdots\in\mathcal{F}\left(\mathcal{S}\right) $ is countable collection of disjoint events, then $ P\left(\cup_{i=0}^{\infty}A_{i}\right)=\alpha_{1}P_{1}\left(\cup_{i=0}^{\infty}A_{i}\right)+\alpha_{2}P_{2}\left(\cup_{i=0}^{\infty}A_{i}\right)=\alpha_{1}\sum_{i=1}^{\infty}P_{1}\left(A_{i}\right)+\alpha_{2}\sum_{i=1}^{\infty}P_{2}\left(A_{i}\right) $$ =\sum_{i=1}^{\infty}\left\{ \alpha_{1}P_{1}\left(A_{i}\right)+\alpha_{2}P_{2}\left(A_{i}\right)\right\} =\sum_{i=1}^{\infty}P\left(A_{i}\right). $
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.)
Note
This problem is identical to the problem in the MBR 2004 Spring Final.
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) $ ?
$ E\left[\mathbf{Y}\left(t\right)\right]=E\left[\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds\right]=\frac{1}{T}\int_{t-T}^{t}E\left[\mathbf{X}\left(s\right)\right]ds=\frac{1}{T}\int_{t-T}^{t}0ds=0. $
(b)
What is the autocorrelation function of $ \mathbf{Y}\left(t\right) $ ?
$ R_{\mathbf{YY}}\left(t_{1},t_{2}\right)=E\left[\mathbf{Y}\left(t_{1}\right)\mathbf{Y}^{*}\left(t_{2}\right)\right]=E\left[\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?
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