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Random Variables Example Problems Solutions

# Random Variables Example Problems Solutions - Random...

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Random Variables – Examples (with solutions) 1. Let X be the number of televisions in an apartment, to be randomly selected in a small town. Suppose that X has probability mass function: x 0 1 2 p ( x ) 0.2 0.7 0.1 (a) Compute the mean/expected value of X . Solution: Let μ X be the mean of X , using the formula for discrete random variables: μ X = E ( X ) = X x xp ( x ) = 0 * p (0) + 1 * p (1) + 2 * p (2) = 0 * (0 . 2) + 1 * (0 . 7) + 2 * (0 . 1) = 0 . 9 (b) What is the probability that this apartment will have at most 2 televisions? Solution: P ( X 2) = X x 2 p ( x ) = p (0) + p (1) + p (2) = 0 . 2 + 0 . 7 + 0 . 1 = 1 Notice that it is impossible for realizations of X to be greater than 2. (c) What is the probability that this apartment will have either exactly 0 televisions or exactly 2 televisions? Solution: We are computing the probability of the union of the events: X = 0 and X = 2, both of which are disjoint, thus: P (( X = 0) ( X = 2)) = P ( X = 0) + P ( X = 2). Equivalently, we can write: P (( X = 0) ( X = 2)) = X ( x =0) ( x =2) p ( x ) = p (0) + p (2) = 0 . 2 + 0 . 1 = 0 . 3 1

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2. Let X be the amount of time (in minutes) it will take to complete and hand-in an exam (where the exam must be handed-in no later than 50 minutes from the start). Suppose that X has a probability density function (pdf): f ( x ) = 1 20 if 30 x < 50 0 otherwise (a) Compute the probability this exam will be handed-in before 20 minutes have passed. Solution: Since the area under f over x 20 is zero, we have that: P ( X 20) = 0 (b) Compute the probability this exam will be handed-in when exactly 30 minutes have passed. Solution: The area under f over the point x = 30, is the area of a line, which is zero. P ( X = 30) = 0 In general, if X is a continuous random variable, the probability that it equals any constant is always zero. (c) Compute the probability this exam will be handed in before 32 minutes have passed. Solution: The area under f over x 32 is equal to the area under f over 30 x 32, which is a rectangle of width 2 and height 1/20, hence: P ( X 32) = 2 * (1 / 20) = 1 / 10 3. A 6-sided die is to be rolled independently 10 times. Let X count the number of even numbers we will roll. (a) What is the name of X ’s distribution? (specify parameter(s)).
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Random Variables Example Problems Solutions - Random...

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