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- IF F(x, y) is the value of the joint distribution function of X and Y at (x, y), show that the marginal distribution function of X is given by G(x) = F(x, ∞) for - ∞ <x < ∞ Use this result to find the marginal distribution function of X for the random variable F(x, y) = { (1-e-x2 ) (1- e-y2) for x>0, y> 0 and 0 elsewhereLet X ~ U[0,1] and Y = -βln(1-X). What is the distribution of Y? Justify.Let X and Y have the joint pdff(x,y)=cx^2y,−y≤x≤1.5,0≤y≤1.5f(x,y)=cx2y a) Determine the value of the constant cc b) Give the value of the marginal pdf fX(x)at x = -0.5 c) Give the value of the marginal cdf FX(x)at x = 0.5 d) Give the value of the conditional distribution fX|Y(x|Y=0.5) at x = 0.5
- a) Find:i) the marginal distribution of X and Y. Are X and Y independent?ii) the probability that fewer than two children fatal given that at least one baby car seats in use at the time of the accident.b) Determine the covariance between X and Y .Give your comment on the relationship between baby car seat and child fatality.Let X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.If X has the discrete uniform distribution f(x) = 1k for x = 1, 2, ... , k, show that(a) its mean is μ = k + 12 ;(b) its variance is σ2 = k2 − 112
- Suppose that you enter a fantasy baseball league. Suppose that the 2021 team budget, say , is randomly drawn from a uniform distribution on the interval , where the unit is U.S. million dollars. In addition, suppose that after the value has been observed , the 2022 team budget, say , is randomly drawn from a uniform distribution on the interval . In other words, the 2022 budget is at most as large as the 2021 budget. a) For any given value of x(50<x<350), obtain E[Y|X=x] b) In view of part (a), obtain E[Y|X] c) Atlanta Braves won the 2021 World Series title. Their estimated 2022 payroll is about $130 million. Would your 2022 fantasy baseball budget be on average larger than their 2022 payroll? Explain briefly.If X1, X2, ... , Xk have the multinomial distribution of Definition 8, show that the mean of the marginal distribu-tion of Xi is nθi for i = 1, 2, ... , k.Let X and Y have the joint pdff(x,y)=cx^2y,−y≤x≤1.5,0≤y≤1.5 a) Determine the value of the constant cc b) Give the value of the marginal pdf fX(x) at x = -0.5 c) Give the value of the marginal cdf FX(x) at x = 0.5 d) Give the value of the conditional distributionfX|Y(x|Y=0.5) at x = 0.5
- Let X1,...,Xn be a random sample from the distribution f(x) = 2x, 0 < x < 1. Find the distribution of the sample maximum X(n)Let f(x) = ½ , -1 < x < 1 0 otherwise be a pdf of the random variable X. Find the distribution function and the pdf of Y= X2The distribution of the random variable X is determined by the function: f(x) = ((x-1)2)/c ; for 1<x<3 = 0 ; inak Sketch the graph of f(x) Determine the FY(y) and fY(y) for Y = X^2 - 1
