homework3

# homework3 - X f 0 x be x x x& ≥ ± = ²< ± ³ Note...

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C OMER ECE 600 Homework 3 1. Let X be a random variable with cdf F X ( x ). Show that if b>a, then F X (b) F X (a). 2. Which of the following are valid cumulative distribution functions? a. ± ² ³ < > = 0 0 1 0 1 1 ) ( F 2 X x x x x x b. ± ² ³ > = c x c x x 1 0 ) ( F X c. ± ² ³ < < - = 2 0 3 2 1 3 1 ) ( F X x x x x x d. x e x - - = 1 ) ( F X e. ) u( ) 1 ( ) ( F X x e x x - - = where c is a constant and u( x) is the unit step function. 3. When ten fair coins are flipped, the event of interest is the number of heads. Let this number be a random variable. Assuming the ten flips are independent of each other, a. Plot the probability mass function for this random variable. b. What is the probability that the random variable is between six and nine inclusive? c. What is the probability that the random variable is greater than or equal to eight?

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4. The median of the random variable X is defined as the real number x m such that P(X x m )=0.5. Find the median of the random variable which has the probability density function
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Unformatted text preview: X , f ( ) 0, x be x x x-& ≥ ± = ² < ± ³ Note: The first step is to find the value of b . 5. Let X be an exponential random variable with parameter ± . a. For x > 0, find P({X & x }). b. For x 2 > x 1 > 0, find P({ x 1 & X & x 2 }). c. For x > 0, find P({X ² x }). d. Segment the positive real line into three equiprobable disjoint intervals. 6. The random variable X is N(10,1). Find f X ( x | (X-10) 2 <4). 7. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by ± ³ ± ² & ≤ > = 10 10 10 ) f( 2 x x x x What is the probability that of 6 such devices at least 3 will function for at least 15 hours? You may assume that the functioning of each device is independent of the functioning of the other devices....
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homework3 - X f 0 x be x x x& ≥ ± = ²< ± ³ Note...

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