The probability density function of a distribution is given byxf(x) = exp(-7).Use differentiation to show that the probability density function has a maximum at x = 0.The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1],q = 1 - p and n is a positive integer. Use the moment generating function to find themean and variance of the distribution in terms of p, q and n.

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Answered: The probability density function of a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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The probability density function of a random variable is kVx+ – 1 f(x) if 1< x< 2 = 0 otherwise Find the k. The marks obtained in mathematics by 1000 students are normally distributed with mean 78% and standard deviation 11%. Determine what was the highest mark obtaianed by the lowest 10% of the student? oninte The probability density function of a random variable is kx³ f(x) if 0sx<2 V25 – x² = 0 otherwise Find the k.
Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 43 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.50 ml/kg for the distribution of blood plasma. (a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.) lower limit 1 upper limit 2 margin of error 3 (b) What conditions are necessary for your calculations? (Select all that apply.) n is large the distribution of weights is normal σ is known the distribution of weights is uniform σ is unknown (c) Interpret your results in the context of this problem. 1% of the intervals created using…
normal distribution has mean= 250 and sd = 20. (a) Find the z score corresponding to x = 235. (b) Find the z score corresponding to x = 320.
A model for normal human body temperature, X, when measured orally in °F, is that it is normally distributed, X ~ N (98.2,0.5184). (i) According to the model, what proportion of people have a normal body temperature of 99 °F or more? (ii) Find the normal body temperature such that, according to the model, only 10% of people have a lower normal body temperature. (iii) Let W denote normal human body temperature, when measured orally in °C. Given that W = (X - 32) and X ~ N(98.2,0.5184), what is the distribution of W? (iv) According to the model that you just derived for W, what proportion of people have a normal body temperature of between 36 °C and 36.8 °C?
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The normal probability density function g(x) = ov2n Where u = mean, o = standard deviation, n = 3.14159, e 2.71828 Let's study a special case where u = 0 and o = 1, which indicates a standard normal probability density function. For simplicity, we scale the function so as to remove the factor 1/(ov2n). Now, we have a new function f(x) f(x) = e (a) Find f'(x) and show the intervals of increase or decrease for f. (b) Find the extreme value of f. (c) Find the intervals of concavity and the inflection points. (d) Sketch the graph of f and point out the extremum value and inflection points.
Use the given probability density function over the indicated interval to find the mean, variance, and standard deviation of the random variable. Then sketch the graph of the density function and locate the mean on the graph.
The probability density function of the net weight in pounds of a packaged chemical herbicide is f(x) = 2.0 for 49.75 < x < 50.25 pounds. Determine the probability that a package weighs more than 50 pounds. i
الموافق foo an exponentiaL distribution fx Cx jt) e.w. where u>o Write (Dankderive) meancespedoed V9lue) f.eECx)
The diameter of a particle of contamination (in micrometers) is modeled with the probability density function 2 f(x)= = x3 for x > 1 . What is the probability that the diameter of the particle of contamination is below 2 micrometers?
The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean 0.62. a) find the P(10<x<15) (10 and 15 are seconds) b) find the P(x>15| x>10) (10 and 15 are seconds) c) find the P(x<15) (15 is seconds)

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