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STA4321HW4

# STA4321HW4 - M X t and use it to ﬁnd E X and V X STA 4321...

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STA 4321 I ntroduction to P robability T heory Assignment 4 Assigned Monday March 15 Due Monday March 22 Show your work to receive full credit. 1. A machine for filling cereal boxes has a standard deviation of 1 ounce in fill per box. Assume that the ounces of fill per box are normally distributed. (a) What setting of the mean ounces of fill per box will allow 14-ounce boxes to overflow only 1% of the time? (b) The company advertises that the box holds 12.8 onces of cereal. Using the mean determined in part (a), what is the probability that a randomly selected box will have less that 12.8 ounces of cereal? [Hint: Can use the standard normal, N (0 , 1), distribution table to get probabilities for any normal random variable X N ( μ, σ 2 ) by “standardizing”, e.g. P ( a < X < b ) = P ( a - μ σ < X - μ σ < b - μ σ ) = P ( a - μ σ < Z < b - μ σ ) where Z N (0 , 1).] 2. (Memoryless Property). Show that if X Exponential( θ ) and t, h > 0, then P ( X > t + h | X > t ) = P ( X > h ) . 3. Let X Gamma( α, β ). Derive the moment generating function,

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Unformatted text preview: M X ( t ), and use it to ﬁnd E( X ) and V( X ). STA 4321 - Introduction to Probability Theory Spring 2010 1 Jorge Rom´an STA 4321 4. The proportion of impurities per batch in a certain type of industrial chemical is a random variable X that has the probability density function f ( x ) = 20 x 3 (1-x ) when 0 ≤ x ≤ 1 and f ( x ) = 0 otherwise. (a) Find the c.d.f of X . (b) Suppose that a batch with a proportion of impurities over 30% cannot be sold. What is the probability that a randomly selected batch cannot be sold for this reason? (c) Find E( X ) and V( X ). [Hint: Is this p.d.f a special case of a known family of p.d.f’s?] (d) Suppose that the dollar value of each batch is given by D = 10-. 75 X . Find E ( D ) and V ( D ). STA 4321 - Introduction to Probability Theory Spring 2010 2 Jorge Rom´an...
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