If øx(@) is a characteristic function of a random variable X, then the nh moment of X is given by m, = (-j)" do" lo=0
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Prove the following
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.1. Define a random variable X as the proportion of time that a student spends on a certain test. Research suggests that the pdf of x is f(x) = (θ + 1)xθ , 0 ≤ x ≤ 1. For the parameter θ, we know θ > −1, but its exact value is still unknown. To estimate the parameter, a random sample of 10 students is obtained, and their proportions of time are: 0.92, 0.79, 0.90, 0.65, 0.86, 0.47, 0.73, 0.97, 0.94, 0.77. Find the Maximum Likelihood Estimator (MLE) of θ.Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
- Consider a random process which is given by Y(t) = t - Z where Z is a random variable with mean 1.2 and second moment 2.5. The autocovariance of the random process X(t) isA simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.A random variable follows a distribution of the form f(x)=k(x+2)e^-x over x>orequalto 0 . Determine the probability that two independent samples are drawn from the population and they both have that 1 < x < 2. State your answer in exact form (with a bunch of e’s), showing all work.
- 13) Random variables X and Y have joint pdf fXY={4xy, 0≤x≤1, 0≤y≤1fXY={4xy, 0≤x≤1, 0≤y≤1 Find Correlation and Covariance2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
- Show that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .X is an exponential random variable with parameter λ = 4 Calculate P {X ≤ 31 Expectation of a function of random variable: Find E(πcosπx) for X uniformly distributed on [0,1].
