Chebyshev's Inequality : P(
STAT 211
Handout 4 (Chapter 4): Continuous Random Variables
A r.v. X is said to be continuous if its set of possible values is an entire interval of numbers. Then a probability distribution of X is f(x) (pdf: probability density function) such that for any two numbers a and b, [pic]
Conditions to be a legitimate probability distribution:
i) f(x) ( 0, for all x
ii) [pic] (area under the entire graph of f(x)).
If X is a continuous r.v., then for any number c, P(X=c)=0.
Cumulative Distribution Function for X (cdf) : F(x)=[pic]
F(-()=0, F(()=1, [pic], P(X>a)=P(X(a)=1-F(a)
Example 1: A college professor never finishes his lecture before the bell rings to end the period and always finishes his lectures within 2 min after the bell rings. Let X=the time that elapses between the bell and the end of the lecture and suppose the pdf of X is [pic]. Find the value of k which makes f(x) a legitimate pdf. (Answer:3/8)
Example 2: The amount of bread (in hundreds of pounds) that a certain bakery is able to sell in a day is a random variable with probability function,
[pic]
a) Find the value of A which makes f(x) a legitimate pdf.
[pic]. Then A=1/25
b) What is the probability that the number of pounds of bread that will be sold tomorrow is
i) more than 500 pounds
P(X>5)= [pic]
ii) less than 500 pounds
P(X ................
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