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Unformatted text preview: ORIE 3510 – Homework 1 (Introduction to Probability Theory) Instructor: Mark E. Lewis due January 30, 2012 (ORIE Hallway drop box) This homework assignment is designed to give you some practice in probability. Some of the concepts should be review, others are leading you to what we will cover in class. Some problems may appear as examples in your text...you still need to do them. 1. A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from the box. (a) What is the sample space? (b) If, at all times, each marble in the box is equally likely to be selected, what is the probability of each point in the sample space? (c) Suppose now that instead of stopping after the second marble, the processing of choosing and replacing a marble continues. Letting X n be the outcome of the n th draw. Provide an example of a sample path of this stochastic process. 2. Suppose X has density function f ( x ) = ( c ( x (2 x )) 0 < x < 2 , otherwise. (a) Compute c . (b) Compute the expected value of X (denoted E X ) (c) Compute the Variance of X (denoted V ar ( X ) = σ 2 ) (d) Compute the standard deviation of X (denoted σ ( X )) 3. Suppose X is exponential with rate λ . (a) Find E e tX for λ > t . (b) Differentiate to find the mean and variance of X . (no credit will be given for no work being shown) (c) Suppose { X 1 ,X 2 ,...,X n } are independent and all exponential with rate λ . Compute E e tY , where Y = X 1 + X 2 + ··· + X n ....
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