Random Variables and Stochastic Processes – Dr. Ghazi Al Sukkar Office Hours: will be posted soon Course Website: 1 Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul /
Cumulative Distribution Function (CDF) 2
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Properties of CDF 4
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Discrete-Type RV 6
Continuous-Type RV 7
Mixed RV. 8
Percentiles 9
Example The distribution function of a random variable X is given by Compute the following quantities : (a) P(X < 2) (b) P(X = 2) (c) P(1 X < 3) (d) P(X > 3/2) (e) P(X = 5/2) (f) P(2<X 7) 10
Example For the experiment of flipping a fair coin twice, let X be the number of tails and calculate F(t), the distribution function of X, and then sketch its graph. Sol : 11
Example Suppose that a bus arrives at a station every day between 10:00 Am and 10:30 AM, at random. Let X be the arrival time; find the distribution function of X, F(t), and then sketch its graph. Sol : 12
Example The sales of a convenience store on a randomly selected day are X thousand dollars, where X is a random variable with a distribution function of the following form : Suppose that this convenience store’s total sales on any given day are less than $2000. (a)Find the value of k. (b)Let A and B be the events that tomorrow the store’s total sales are between 500 and 1500 dollars, and over 1000 dollars, respectively. Find P(A) and P(B). (c)Are A and B independent events? 13
Probability Density Function (pdf) 14
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Pdf for Discrete Random Variables 17
Example In the experiment of rolling a balanced die twice, let X be the maximum of the two numbers obtained. Determine and sketch the probability mass function and the distribution function of X. Sol : 18
Example Can a function of the form be a probability mass function ? Sol : 19
Example Let X be the number of births in a hospital until the first girl born. Determine the probability mass function and the distribution function of X. Assume that the probability is 1/2 that a baby born is a girl. Sol : 20