Consider a random variable X ∼ exp(1) and define Y = 1{X>1}. Find the cumulative distribution function of (X, Y ).
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Consider a random variable X ∼ exp(1) and define Y = 1{X>1}. Find the cumulative distribution
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- Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?Show that if X is a random variable with continuous cumulative distribution function F(x), then F(x)=U is uniformly distributed over the interval (0,1).The probability function of the random variable X is defined as f(x)=cx²(1-x)² for 0<x<1, otherwise f(x)= 0. Calculate the constant c , the expected value and the variance.
- Find the variance by calculating the first two moments of the random variable X = (- 1 / λ) ln (1-U), where U ~ U (0,1) and λ> 0.Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.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-Y2