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Problem 4. Consider a continuous random variable X with the density function (called the exponential density) f ( x ) = e x , x >= 0 . 1. Find and sketch the cdf for x . 2. Find the mean and variance of X . (Hint: Use integration by parts.) 3. Find P ( 1 <= X <= 2 ) . (1) general CDF formula: x e x d x = 0 x f ( t ) d t [ e x ] 0 x = ( e x ) ( e 0 ) = e x + 1 = 1 e x .

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- 1. Problem 4. Consider a continuous random variable X with the density function (called the exponential density) f ( x ) = e x , x >= 0 . 1. Find and sketch the cdf for x . 2. Find the mean and variance of X . (Hint: Use integration by parts.) 3. Find P ( 1 <= X <= 2 ) . (1) general CDF formula: x e x d x = 0 x f ( t ) d t [ e x ] 0 x = ( e x ) ( e 0 ) = e x + 1 = 1 e x