.Define gamma distribution with parameters 5 and 7 and obtain its moment generating function. Hence or otherwise find its mean and variance.
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.2.2 Explain how to generate from the Beta (1, 3) distribution using the inverse- transform method.(13) Let X b(6,-) find E(5+6x) and distribution function. 3
- Suppose that Y has an exponential distribution with mean Beta. Show that 2Y/Beta has a chi square distribution with 2 degrees of freedom.23. Find the cumulant generating function of gamma distribution and obtain various cumulants.4) Considey molecules cnd 100 10 cells of equal energy. Find log. energy most probabile and ii the wfor j the distribution least probable distribution.
- Let X have a Poisson distribution. If , find the mode of the distribution. Suppose Y has a uniform pdf over the interval [3,5]. Find its (1) first quartile, its (2) median, and (3) the sum of its third quartile and its ninth decile.Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.
- Let X∼Uniform(0,1) distribution. Find the PDF of Y= 3√X and derive the expected value and the variance of Y.Find the PSD of a stationary random process for which Auto correlation is Rx (t)= 6- e alti5. Find the moment generating function of the exponentially distributed random vari- able with pdf Xe-, x≥0 and use it to evaluate the mean and variance of X.