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Suppose X is a continuous random variable with cdf F, where F is invertible with inverse function F$^{-1}$. Let U~Unif(0, 1). Then the distribution of F$^{-1}$(U) is equal to the distribution of X. This means that X can be simulated as F$^{-1}$(u) where u is a simulated value from the Unif(0,1) distribution. 1. Consider the continuous random variable X with pdf given by $f_X(x) = \begin{cases} 2e^{-2x}, & x \ge 0\\ 0, & \text{otherwise} \end{cases}$ (a) What is the cdf of X? (b) Find E(X). Be sure to show your steps. (c) Determine F$^{-1}$(U) and use R to simulate 10,000 observations from this pdf by the inverse transform method. (d) Compute the mean of the generated observations and compare the result with E(X) that was obtained in part (b). 2. Suppose Y has pdf $f_Y(y) = \begin{cases} c(2 - y), & 0 \le y \le 2\\ 0, & \text{otherwise} \end{cases}$. For this distribution, find each of the following. Be sure to show your steps. (a) c (b) F(y) (c) P(1 < Y ? 2) (d) Mean of Y (e) Variance of Y

          Suppose X is a continuous random variable with cdf F, where F is invertible with inverse function F$^{-1}$.
Let U~Unif(0, 1). Then the distribution of F$^{-1}$(U) is equal to the distribution of X. This means that X
can be simulated as F$^{-1}$(u) where u is a simulated value from the Unif(0,1) distribution.
1. Consider the continuous random variable X with pdf given by $f_X(x) = \begin{cases} 2e^{-2x}, & x \ge 0\\ 0, & \text{otherwise} \end{cases}$
(a) What is the cdf of X?
(b) Find E(X). Be sure to show your steps.
(c) Determine F$^{-1}$(U) and use R to simulate 10,000 observations from this pdf by the inverse
transform method.
(d) Compute the mean of the generated observations and compare the result with E(X) that was
obtained in part (b).
2. Suppose Y has pdf $f_Y(y) = \begin{cases} c(2 - y), & 0 \le y \le 2\\ 0, & \text{otherwise} \end{cases}$. For this distribution, find each of the
following. Be sure to show your steps.
(a) c
(b) F(y)
(c) P(1 < Y ? 2)
(d) Mean of Y
(e) Variance of Y

Suppose X is a continuous random variable with cdf F, where F is invertible with inverse function F^-1.
Let U Unif(0, 1). Then the distribution of F^-1(U) is equal to the distribution of X. This means that X
can be simulated as F^-1(u) where u is a simulated value from the Unif(0,1) distribution.
1. Consider the continuous random variable X with pdf given by fX(x) = 2e^-2x, x ≥ 0
0, otherwise
(a) What is the cdf of X?
(b) Find E(X). Be sure to show your steps.
(c) Determine F^-1(U) and use R to simulate 10,000 observations from this pdf by the inverse
transform method.
(d) Compute the mean of the generated observations and compare the result with E(X) that was
obtained in part (b).
2. Suppose Y has pdf fY(y) = c(2 - y), 0 ≤ y ≤ 2
0, otherwise. For this distribution, find each of the
following. Be sure to show your steps.
(a) c
(b) F(y)
(c) P(1 < Y ? 2)
(d) Mean of Y
(e) Variance of Y

Added by Brian B.

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