find mean and variance ,MGF of one random variable derive that step by step for number 2,3,4.Thank you
3:
Hence, MGF is
Differentiating with respect to t gives:
The mean is:
Differentiating with respect to t again gives:
So,
The variance is:
4:
Here we will use the intergral
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PDF of gamma distribution with parameter and is
Let us assume
.
So pdf of gamma distribution will be
So MGF will be
So MGF is
Now putting
gives
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Differentiating MGF with respect to t once gives:
Noe putting t=0 in the above equation gives:
Differentiating MGF with respect to t again gives:
Noe putting t=0 in the above equation gives:
Now putting
gives
Therefore variance is:
find mean and variance ,MGF of one random variable derive that step by step for number...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
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Given the common probability distributions & moment generating functions NOTES Is very desirable to be used in applications but both the PDF MGF Normal ALT/ Notation N(μ,σ) population mean μ and population st dev σ sample space Ω is defined or all X must be known. s an approximation to the normal dist for smaller samples, with degrees freedom v T dist N/A ALT/ Notation T PLEASE LET ME KNOW IF YOU FIND AN MGF Sample space is defined Forx>0...
Q2. More about operations with expectation and covariances Recall that the variance of random variable X is defined as Var(X) Ξ E 1(X-E(X))2」, the covariance is Cor(X, Y-E (X-E(X))(Y-E(Y)), and the correlation is Corr(X,Y) Ξ (a) What is the value of EX-E(X))? (Hint: Let μ denote E(X). Then, the parameter μ is a unknown, but fixed value like a constant.) (0.5 pt) b) The following is the proof that Var(X) E(X2) E(X)2: -E(x)-E(x)2 In a similar way, prove that Cov(X,...