Theorem 1.6 (The Kolmogorov inequality) Let X1, X2, Xn be independent random variables with mean 0 and suppose that Var Xk<∞ for all k. Then, for x > 0, P(max Sk>x) ≤ Isk≤n Σ-Var X In particular, if X1, X2,..., X, are identically distributed, then P(max Sx) ≤ Isk≤n nVar X₁ x2
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- Suppose X is random variable whose p.d.f. is f2)=(2x-x²), 0, OSxs2 Suppose X is random variable whose p.d.f. is f(x)={4 elsewhere Find the mode , if it exists.An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.Exercise 9. Suppose X1, X2, , Xn are independent random variables with X; ~ T(ai, B), 1Solve subpart (iv). i-iii already solvedSuppose X, Y and Z are independent random variables with var (X) = 4 , var (Y) = 4 , var (Z) = 1/n Find: var (X/2 - Y + the square root of nZ) (divided by -2)Suppose that X₁, X2, X3 are independent and identically distributed random variables with distribution function: Fx (x)=12* for x ≥ 0 and Fx (x) = 0 for x 4).When X is a binomial random variable, then the probability X ≤ r successes can be found using the formula or the calculator command: binompdf(n, p, r). P(X < r) = binomcdf(n,p,r-1) P(X > r) = 1 - P(X ≤ r ) = 1 - binomcdf(n,p,r) P(X ≥ r) = 1 - P(X < r) = 1 - binomcdf(n,p,r-1) If X is a binomial random variable, where there are 175 trials, and p = 0.057 find P(X ≥ 2)When X is a binomial random variable, then the probability X ≤ r successes can be found using the formula or the calculator command: binompdf(n, p, r). P(X < r) = binomcdf(n,p,r-1) P(X > r) = 1 - P(X ≤ r ) = 1 - binomcdf(n,p,r) P(X ≥ r) = 1 - P(X < r) = 1 - binomcdf(n,p,r-1) If X is a binomial random variable, where there are 125 trials, and p = 0.016 find P(X < 3)Q3) A) A grocery store sells X hundred kilograms of rice every day, where the distribution of the random variable X is of the following form:. x < 0 0Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 24 psi. Suppose the actual air pressure in each tire is a random variable-X for the right tire and Y for the left tire, with joint pdf will be f(x, y) = K (x² + y²) if 20 ≤ x ≤ 30, 20 ≤ y ≤ 30 Are X and Y independent random variables? In order to answer this question, you need to find the value for K first. X and Y are not independent. X and Y are independent. cannot decide none of the other option is correctLet X1 , X2 , X3 be a collection of independent discrete random variables that all take the value 1 with probability p and take the value 0 with probability (1-p). The mean and variance of p-hat = 1/n ( X1 + X2 +... Xn ) . E(p-hat)= p Var(p-hat) = p(1-p) / n The parts to this question are attached3.) Find P (X > Y) where X and Y are independent random variables that satisfy X~ N (2,1) and Y~ N (6,3)SEE MORE QUESTIONSRecommended textbooks for youAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage