Let X be a random variable with continuous cdf F. Show that U = F(X) Problem 4 s uniformly distributed over [0, 1], i.e. U - Uniform(0, 1).
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- Problem 7: Let X be a continuous random variable with the probability density for f(x) = 3x2 values of x in [0,1], and f(x) = 0 elsewhere. Compute the expected value and variance of X.Problem 1. A continuous random variable X is defined by f(x)=(3+x)^2/16 -3 ≤ x ≤ -1 =(6-2x^2)/16 -1 ≤ x ≤ 1 =(3-x^2)/16 -1 ≤ x ≤ 3 a)Verify that f(x) is density. b)Find the MeanConsider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?
- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2If X and Y are random variables and X is a geometric random variable where p = 0.1 then what is the probability mass function of Y = sin(X*pi) ?The density of a random variable X is f(x) = C/x^2 when x ≥ 10 and 0 otherwise. Find P(X > 20).
- X is a continuous random variable taking values between 0 and 2. If F(1.5) = 0.70 and P(X<1) = 0.20, what is P (1 < X < 1.5)?Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0If two random variables X and Y are independent with marginal pdfs fx(x)= 2x, 0≤x≤1 and fy(y)= 1, 0≤y≤1 Calculate P(Y/X>2)
- If X is a continuous random variable, then we know P(a≤x≤b) is equal to the area under PDF curve between a and b. So, if k is some particluar value of X, then what is the value of P(X=k)Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)The joint PDF of the random variables X and Y is constant on the shaded region, as shown in the image below, and is zero outside. It can be determined that fX,Y(x,y)=2/3. Determine E[XY]. (The answer is not 3/4)