Continuous random variable and Probability density function
Quantities whose values range over an interval of numbers are called continuous random variable denoted by X
That is, if for some A<B, any number x between A and B is possible.
Let X be a continuous rv. Then a pdf of X is a function f(x) such that for any two numbers a and b with ab,
For f(x) to be a legitimate pdf, it must satisfy the following two conditions
A continous rv X is said to be uniform distribution on the interval [A, B] if the pdf of X is
“Time Headway” in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. The following pdf of X is essentially the one suggested in “The statistical properties of freeway traffic” (Transportation Research Vol 11, 221-228)
Show that the above expression is a legitimate pdf.
There is no headway associated with headway times less than 0.5 and headway density decreases rapidly as x increases from 0.5. clearly
To show that , we use the calculus result
In commuting to work, I must first get on a bus near my house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A=0, and B =5, then it can be shown that my total weighting time Y has a pdf
Sketch a graph of the pdf of Y
What is the probability that total waiting time is at most 3 min
What is the probability that total waiting time is at most 8 min
What is the probability that total waiting time is between 3 and 8 min
What is the probability that total waiting time is either less than 2 min or more than 6 min?
Cumulative distribution function
The cumulative distribution function F(x) for a continuous rv X is defined for every number x by
F(X)= P(X≤ x ) =
Using F(x) to compute probabilities
Let X be a continuous rv with pdf f(x) and cdf F(x). Then for any number a,
P(X > a )=1- F(a)
and for any two numbers a and b with a < b,
P(a ≤ X ≤ b )= F(b) - F(a)
Expected Values for Continuous Random Variables
The expected or mean value of a continuous rv X with pdf f(x) is
The variance of a continuous random variable X with pdf f(x) and mean value µ is
THE NORMAL DISTRIBUTION
A continuous rv X is said to have a normal distribution with parameters µ and σ (or µ and σ2) , where and , if the pdf of X is
The statement that X is normally distributed with parameters and 2 is often abbreviated
The value of is the distance from to the inflection point of the curve (the point at which the curve changes from turning downward to turning upward). Large value of yield graphs that are quite spread about , whereas small values of yields graphs with a high peak above and most of the area under the graph quite close to . Thus a large implies that a value of x far from may well be observed, whereas such a value is quite unlikely when is small.
Important Point: None of the standard integration techniques can be used to evaluate the above expression, instead, for =0, and =1.
Standard Normal Distribution
The normal distribution with parameter values =0, =1 is called the standard normal distribution. A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by Z. The pdf of Z is
The cdf of Z is P(Z ≤ z) = , which will be denote by
When , probabilities involving X are computed by “standardizing”. The standardized variable is . Subtracting shifts the mean from to zero, and then dividing by scales the variable so that the standard deviation is 1 rather than .
If X has a normal distribution with mean and standard deviation , then
, has a standard normal distribution. Thus
The key idea behind the standardizing is that any probability involving X can be expressed as a probability involving a standard normal rv Z, so that standard normal tables can be used.
More Illustration through diagram, about the equality of nonstandard and standard normal distribution curve
Illustration of Standard Normal Tables
Important Properties if the population distribution is normal
68% of the values are within 1 SD of the mean
95% of the values are within 2 SD of the mean
99.7% of the values are within 3 SD of the mean.
1. In each case, determine the value of constant c that makes the probability statement correct.
2. The average grade for an exam is 74, and the standard deviation is 7. If 12% of the class are given A’s and the grades are curved to follow a normal distribution, what is the lowest possible A and the highest possible B?
3. A professional commutes daily from his suburban home to his midtown office. The average time of a one-way trip is 24 min, with a stand. deviation of 3.8 mins. Assume the distribution of trip times to be normally distributed.
a) What is the probability that a trip will take atleast ½ hour.
b) If the office opens at 9:00 AM and he leaves his house at 8:45 AM daily, what percentage of the time is he late for work?
c) If he leaves the house at 8:35AM, and coffee is served at the office from 8:50 AM until 9:00 AM, what is the probability that he misses coffee?
d) Find the length of the time above which we find the slowest 15% of the trips.
e) Find the probability that 2 of the next 3 trips will take at least ½ hour.
The GAMMA Distribution
The graph of any normal Pdf is bell-shaped and thus symmetric. There are many practical situations in which the variable of interest to the experimenter might have a skewed distribution. A family of pdf’s that yields a wide variety of skewed distributional shapes is the gamma family.
e.g. Monthly income distribution of the population
To define the family of gamma distributions, we first need to introduce a function that plays an important role in many branches of mathematics.
For the gamma function is defined by
The most important properties of gamma function are the following:
For any positive integer n,
A continous rv X is said to have a gamma distribution, if the pdf of X is
= Shape factor
Special cases of the gamma distribution include:
When the shape parameter is set to one, and the scale parameter to the mean, the gamma distribution simplifies to the exponential.
A chi-squared distribution is a gamma distribution in which the shape parameter set to the degrees of freedom divided by two and the scale parameter set to two.
Few more distributions: Weibull Distribution, Lognormal Distribution And Beta Distribution
A rv X is said to have a Weibull distribution with parameters and
A non-negative rv X is said to have a lognormal distribution if the rv Y=ln(X) has a normal distribution. The resulting pdf of a lognormal rv when ln(X) is normally distributed with parameters and is
The parameters and are not the mean and standard deviation of X but of ln(X).
All families of continuous distributions discussed so far except for the uniform distribution have positive density over an infinite interval. The Beta distribution provides density only for X in an interval of finite length.
A random variable X is said to have a beta distribution with parameter and (both positive), A, and B if the pdf of X is