The cumulative distribution function of the random variable X is given by F(x) = 1-e-r' (z...
9. The distribution function of a random variable X is given by 0, for r<-1, F(x) = { 271 -1<x<1, 1, 2 > 1. Find (a) P(Z < X < }); (b) P(1<x< 2).
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
X Y Z iid Suppose for random variable X, P(X > a) - exp( random variable Y, P(Y > y) exp(-0y) for y > 0, and for random variable , P(Z > z)--exp(-фа) for z > 0. (a) Obtain the moment generating functions of X, Y and Z. (b) Evaluate E(X2IX > 1) and show it is equal to a quadratic function of λ. (c) Calculate P(X > Y Z) if λ-1, θ--2 and φ--3. -λα) for x > 0,...
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function F and density f. Let b>0. (a) Write the forinula for E(X b)+1. (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> 0. (a) Find the cumulative distribution function of Y = XI(X < b} (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx and density fx, and let c>O. Verify that for the Value-at-Risk we have VaR,x (p) = cVaRx (p)
Previous Problem: Determine values of the cumulative distribution function for the random variable in the previous problem. 3. 2. The probability mass function below is defined for x 0, 1,2,3,.. fr 5 5 -56 What is the probability for each of the following expressions? a) P(X 2) b) P(XE 2) c) P(X> 2) d) P(X2 1)
Q3. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b>0 (a) Find the cumulative distribution function of Y- (X -b)+ (b) Apply the general formula from (a) to Pareto distribution with parameter a > 0. Hint: Consider separately cases b e (0, 1] and b> 1.
1. Suppose the random variable as a uniform distribution on [-k,k] a) Construct the pdf X. b) Calculate P(X > 2X > 1) in terms of 'k c) Calculate 'K if P(-2 < X < 2) =