(a) Show that the moment generating function is MX 1t 2 et t1 et 2 in .NET Attach QR Code in .NET (a) Show that the moment generating function is MX 1t 2 et t1 et 2 (a) Show that the moment generating function is MX 1t 2 et t1 et 2VS .NET qr code generator with .netgenerate, create qr code none in .net projects(b) Find the mean and variance of X. S5-21. Let X1, X2, . . . , Xr be independent exponential random variables with parameter . (a) Find the moment generating function of Y X1 X2 p Xr. (b) What is the distribution of the random variable Y [Hint: Use the results of Exercise S5-20]. S5-22. Suppose that Xi has a normal distribution with mean 2 1, 2. Let X1 and X2 be independent. i and variance i,i (a) Find the moment generating function of Y X1 X2. (b) What is the distribution of the random variable Y S5-23. Show that the moment generating function of the chi-squared random variable with k degrees of freedom is MX (t) (1 2t) k 2. Show that the mean and variance of this random variable are k and 2k, respectively. S5-24. Continuation of Exercise S5-20. (a) Show that by expanding etX in a power series and taking expectations term by term we may write the moment generating function as MX 1t 2 E 1etX 2 1 r 1 t tr r! p 2scan qr code jis x 0510 for .netUsing Barcode reader for .NET Control to read, scan read, scan image in .NET applications.(b) Use MX (t) to nd the mean and variance of X. S5-19. A random variable X has the exponential distribution f 1x2 eBarcode barcode library in .netusing .net toreceive bar code in asp.net web,windows applicationt2 2!recognize barcode on .netUsing Barcode reader for .net vs 2010 Control to read, scan read, scan image in .net vs 2010 applications.(a) Show that the moment generating function of X is MX 1t 2 a1 t b Control qr size on visual c# qr code jis x 0510 size for c#(b) Find the mean and variance of X. S5-20. A random variable X has the gamma distribution Control qr codes image on .netusing barcode writer for aspx.cs page control to generate, create qr barcode image in aspx.cs page applications.Thus, the coef cient of t r r! in this expansion is r , the rth origin moment. (b) Continuation of Exercise S5-20. Write the power series expansion for MX(t), the gamma random variable. (c) Continuation of Exercise S5-20. Find 1 and using the 2 results of parts (a) and (b). Does this approach give the same answers that you found for the mean and variance of the gamma random variable in Exercise S5-20 Control quick response code size in visual basic.net qr code jis x 0510 size for vb.net5-10EAN / UCC - 14 drawer with .netusing .net vs 2010 crystal toembed uss-128 with asp.net web,windows applicationCHEBYSHEV S INEQUALITY (CD ONLY)Visual .net matrix barcode integrating for .netusing visual .net toinclude matrix barcode in asp.net web,windows applicationIn 3 we showed that if X is a normal random variable with mean and standard deviation , P( 1.96 < X < 1.96 ) 0.95. This result relates the probability of a normal random variable to the magnitude of the standard deviation. An interesting, similar result that applies to any discrete or continuous random variable was developed by the mathematician Chebyshev in 1867.Connect qr code iso/iec18004 in .netuse .net framework crystal qr code encoding tomake qr-codes on .net5-14Visual Studio .NET quick response code integrated for .netuse .net vs 2010 qr code generator togenerate quick response code on .netChebyshev's Inequality PLANET barcode library for .netusing barcode integrating for .net vs 2010 crystal control to generate, create usps confirm service barcode image in .net vs 2010 crystal applications.For any random variable X with mean P10 X for c > 0. 0 Control datamatrix image in microsoft worduse word ecc200 implementation todisplay data matrix ecc200 on wordand variance c 2 Visual Studio .NET (WinForms) code 128b implementation on .netuse .net winforms code 128 code set c maker toencode code 128a on .net1 c2Control code-39 size on word documents 3 of 9 barcode size with office wordThis result is interpreted as follows. The probability that a random variable differs from its mean by at least c standard deviations is less than or equal to 1 c2. Note that the rule is useful only for c > 1. For example, using c = 2 implies that the probability that any random variable differs from its mean by at least two standard deviations is no greater than 1 4. We know that for a normal random variable, this probability is less than 0.05. Also, using c 3 implies that the probability that any random variable differs from its mean by at least three standard deviations is no greater than 1 9. Chebyshev s inequality provides a relationship between the standard deviation and the dispersion of the probability distribution of any random variable. The proof is left as an exercise. Table S5-1 compares probabilities computed by Chebyshev s rule to probabilities computed for a normal random variable. EXAMPLE S5-8 The process of drilling holes in printed circuit boards produces diameters with a standard deviation of 0.01 millimeter. How many diameters must be measured so that the probability is at least 8 9 that the average of the measured diameters is within 0.005 of the process mean diameter Let X1, X2, . . . , Xn be the random variables that denote the diameters of n holes. The average measured diameter is X 1X1 X2 p Xn 2 n. Assume that the X s are independent random variables. From Equation 5-40, E1X 2 and V1X 2 0.012 n. Consequently, the 2 1 2 standard deviation of X is (0.01 n) . By applying Chebyshev s inequality to X , P1 0 X Let c = 3. Then, P10 X Therefore, P10 X 0 310.012 n2 1 2 2 8 9 0 310.012 n2 1 2 2 1 9 0 c10.012 n2 1 2 2 1 c2Control code 128a data in visual basic barcode standards 128 data in vb.netTable S5-1 Percentage of Distribution Greater than c Standard Deviations from the Mean c 1.5 2 3 4 Chebyshev s Rule for any Probability Distribution less than 44.4% less than 25.0% less than 11.1% less than 6.3% Normal Distribution 13.4% 4.6% 0.27% 0.01%QR Code ISO/IEC18004 integrated with vb.netgenerate, create qr-code none with visual basic.net projects5-15Control barcode 3 of 9 data for vb.net ansi/aim code 39 data for vbThus, the probability that X is within 3(0.012 n)1 2 of such that 3(0.012 n)1 2 0.005. That is, n EXERCISES FOR SECTION 5-10Control code 128c size for visual basic.netto include code 128a and code 128a data, size, image with vb barcode sdkS5-25. The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Bound the probability that the thickness is less than 6 or greater than 14 micrometers. S5-26. Suppose X has a continuous uniform distribution with range 0 x 10. Use Chebyshev s rule to bound the probability that X differs from its mean by more than two standard deviations and compare to the actual probability. S5-27. Suppose X has an exponential distribution with mean 20. Use Chebyshev s rule to bound the probability that X differs from its mean by more than two standard deviations and by more than three standard deviations and compare to the actual probabilities. S5-28. Suppose X has a Poisson distribution with mean 4. Use Chebyshev s rule to bound the probability that X differs from its mean by more than two standard deviations and by more than three standard deviations and compare to the actual probabilities. S5-29. Consider the process of drilling holes in printed circuits boards. Assume that the standard deviation of the diameters is 0.01 and that the diameters are independent. Suppose that the average of 500 diameters is used to estimate the process mean. (a) The probability is at least 15 16 that the measured average is within some bound of the process mean. What is the bound (b) If it is assumed that the diameters are normally distributed, determine the bound such that the probability is 15 16 that the measured average is closer to the process mean than the bound. S5-30. Prove Chebyshev s rule from the following steps. De ne the random variable Y as follows: Y (a) (b) (c) (d) e 1 0 if 0 X 0 otherwise cControl code-39 size with .net code 39 extended size with .net