3 Continuous RV.

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Presentation transcript:

3 Continuous RV

cdf 3-1: X is the number of heads in three tosses of a coin. Find P[1<X2], P[0.5X<2.5], P[1X<2]. 3-2: Draw the cdf of a Zipf RV X with 3 elements. That is, 𝑃 𝑋=𝑘 = 1 𝑐 𝐿 1 𝑘 for k=1,2,3

cdf vs. pdf 3-3: Find the cdf of an RV X. Its pdf is given by: 3-4: Find the cdf and pdf of the transmission time X of messages in a network. Its probability is given by an exponential distribution:

expectation 3-5: Find E[X-Y], where an RV X has cdf of 1−𝑒 −2𝑥 , and RV Y has the pdf of 3𝑒 −3𝑥 Note that X and Y are non-negative RVs

Exponential RV 3-6: Find the probability that a man who has already waited for a bus for 2 min will wait longer than total 4 min? A bus comes every 2 min on average. Inter-bus arrival times follow an exponential distribution.

Normal RV Look up Table 4.2 3-7: X is a normal RV with N(80,102). Find P[X>90] and P[70<X<90] 3-8: X is a normal RV with N(80,502). Find P[X>90] and P[70<X<90]

Normal RV Look up Table 4.3 3-9: Scores of students are modeled by a normal RV with N(80,102). Find which score is top 10% and top 1%.

Chebyshev Inequality 3-10: if X has mean m and variance 2, Suppose X is a Gaussian RV. What is the probability that P[ |X-m|  2 ]? Use Chebyshev inequality .