Let Z be a discrete random variable with ?(Z)0. Does it necessarily follow that E(Z^3)=0? If yes, give a proof; if no, give a counterexample.
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Let Z be a discrete random variable with ?(Z)0. Does it necessarily follow that E(Z^3)=0? If yes, give a proof; if no, give a counterexample.
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- What should be the value of a to make this continuous random variable X valid?Without using a moment generating function; Prove that the variance of a beta-distributed random variable with parameters α and β is σ2 = αβ/[(α + β)^2 (α + β + 1)]Let us consider a discrete random variable having the pmf given by, PX(k) ={ (1/3) , k = 1 (2/3) , k = 2. 0 , k = 3 Calculate the moment generating function for X. Also get the values of its mean and variance.
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- An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is λ is Poisson distributed with mean λ. They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameters s and α. (a) If a newly insured person has n accidents in her first year, find the conditional density of her accident parameter. (b) Also, determine the expected number of accidents that she will have in the following year.Let X be a continuous random variable with mean μ and standard deviation σ. If X is transformed to Y = 2X + 3, what are the mean and standard deviation of Y?Suppose that the random variable Y has the Uniform(θ,2θ) distribution where θ > 0 is a parameter. Using an appropriate pivotal quantity, verify that if 0 < α (alpha) < 1,then