A random variable [latex]X[/latex] is sampled 36 times obtaining a value [latex]\overline{X}[/latex] that is used as an estimate of [latex]E[X][/latex]. If the PDF of [latex]X[/latex] is [latex]f_X(x)=2e^{-2x}[/latex], where [latex]x\geq 0[/latex], a. Determine and [latex]E[\overline{X}][/latex] and [latex]E\bigg[\overline{X}^2\bigg][/latex]. b. What is the probability that the sample mean lies between [latex]1/4[/latex] and [latex]3/4[/latex]?

a. Since [latex]X[/latex] is an exponential random variable, the true mean and true variance are given by [latex]\,\displaystyle E[X]=1/2[/latex] and [latex]\,\displaystyle \sigma_X ^2 =1/4[/latex]. Thus, [latex]\,\displaystyle E[\overline{X}]=E[X]=1/2[/latex]. Since [latex]n=36[/latex], the second moment of [latex]\overline{X}[/latex] is given by [latex]\,\displaystyle E\left[\overline{X}^2\right]=\sigma_{\overline{X}}^2+\{E[\overline{X}]\}^2=\frac{\sigma_X^2}{n}+\{E[X]\}^2=\frac{1 / 4}{36}+\left (\frac{1}{2}\right )^2=\frac{37}{144}[/latex] b. Since [latex]n=36[/latex], the variance of the sample mean becomes [latex]\,\displaystyle \sigma_{\overline{X}}^2 = \sigma_{{X}}^2 =36=1/144[/latex]. The probability that the sample mean lies between [latex]1/4[/latex] and [latex]3/4[/latex] is given by [latex]\,\displaystyle \begin{aligned} P\left[\frac{1}{4} \leq \overline{X} \leq \frac{3}{4}\right] & =F_{\overline{X}}\left (3 / 4\right )-F_{\overline{X}}\left (1 / 4\right )=\Phi\left (\frac{3 / 4-1 / 2}{1 / 12}\right )-\Phi\left (\frac{1 / 4-1 / 2}{1 / 12}\right ) \\ & =\Phi\left (3\right )-\Phi\left (-3\right )=\Phi\left (3\right )-\{1-\Phi\left (3\right )\}=2 \Phi\left (3\right )-1 \\ & =2\left (0.9987\right )-1=0.9974 \end{aligned}[/latex] where the values used in the last equality are taken from Table 1 of Appendix 1.

Question 9.1: A random variable X is sampled 36 times obtaining a value X ......

Fundamentals of Applied Probability and Random Processes [1392375]

A random variable $X$ is sampled 36 times obtaining a value $\overline{X}$ that is used as an estimate of $E[X]$ . If the PDF of $X$ is $f_X(x)=2e^{-2x}$ , where $x\geq 0$ ,

a. Determine and $E[\overline{X}]$ and $E\bigg[\overline{X}^2\bigg]$ .

b. What is the probability that the sample mean lies between $1/4$ and $3/4$ ?

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a. Since $X$ is an exponential random variable, the true mean and true variance are given by $\,\displaystyle E[X]=1/2$ and $\,\displaystyle \sigma_X ^2 =1/4$ . Thus, $\,\displaystyle E[\overline{X}]=E[X]=1/2$ . Since $n=36$ , the second moment of $\overline{X}$ is given by

\,\displaystyle E\left[\overline{X}^2\right]=\sigma_{\overline{X}}^2+\{E[\overline{X}]\}^2=\frac{\sigma_X^2}{n}+\{E[X]\}^2=\frac{1 / 4}{36}+\left (\frac{1}{2}\right )^2=\frac{37}{144}

b. Since $n=36$ , the variance of the sample mean becomes $\,\displaystyle \sigma_{\overline{X}}^2 = \sigma_{{X}}^2 =36=1/144$ . The probability that the sample mean lies between $1/4$ and $3/4$ is given by

\,\displaystyle \begin{aligned} P\left[\frac{1}{4} \leq \overline{X} \leq \frac{3}{4}\right] & =F_{\overline{X}}\left (3 / 4\right )-F_{\overline{X}}\left (1 / 4\right )=\Phi\left (\frac{3 / 4-1 / 2}{1 / 12}\right )-\Phi\left (\frac{1 / 4-1 / 2}{1 / 12}\right ) \\ & =\Phi\left (3\right )-\Phi\left (-3\right )=\Phi\left (3\right )-\{1-\Phi\left (3\right )\}=2 \Phi\left (3\right )-1 \\ & =2\left (0.9987\right )-1=0.9974 \end{aligned}

where the values used in the last equality are taken from Table 1 of Appendix 1.

Table 1 Area Φ(x) under the standard normal curve to the left of x
x	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974

Table 1 Area Φ(x) under the standard normal curve to the left of x—cont’d
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998	0.9998
3.5	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998
3.6	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.7	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.8	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	1.0000	1.0000	1.0000