Chapter 4 Continuous Distributions

Section 4.1   Probability Densities

Section 4.2   Exponential and Gamma Distributions

Section 4.5  Cummulative Distribution Functions

Continuous random variables. Random variables whose range is an interval of real numbers.

Examples:

Normal Distribution: possible values are any real number

Two other important ones we will look at are Uniform Distribution over an interval ( a, b), and Exponential Distribution.

 

Section 4.1 Probability Densities

• For discrete random variables, we used histograms to diagram probability distributions.
• The sum area of the bars of the histogram represented the probability that a random variable that distribution lies within the specified interval.
• For random variables with a continuous distribution, we use a probability density function f(x) to determine the probability that the random variable lies with an interval as follows:
• A random variable X has probability density function f(x) if for all a <=b

• The interpretation of this probability graphically is the area under the curve f(x) between a and b. ( See page 260)
• The expectation of a continuous random variable X with density f(x) is defined to be

provided the integral exists (i.e. converges absolutely).

(May not exist as in example 3 page 271)

• The expectation of a function g(X) of a continuous random variable X with density f(x) is defined to be

provided the integral exists (i.e. converges absolutely).

• For example E(X²) =
• Study the comparisons on page 262-263 of concepts associated with discrete distributions with continuous distributions.
• Example: Exercise 3 page 275.

a) To find the value of c , we must have
 
so c must be 6

b) Calculate

c) Same as b) except integrate from 0 to 1/3
 

d)  Integrate from 1/3 to 1/2 OR just subtract answer in c) from answer in b)

• Example: Exercise No. 2 page 275

a) To find the value of c , we must have



Note: For continuous random variables we don't calculate P[X = x] since this is always 0. We can say that P[X is in a small interval dx ] is approximately f(x)*dx
 
 

• Independence of continuous random variables is defined the same way as for discrete random variables.

• But, we don't reduce this to individual values like we do for discrete random variables --

i.e. we CAN'T SAY



since each of these is always 0 for continuous distributions.

Important Continuous Distributions

The Uniform Distribution on the interval (a,b)

• A random variable X with the uniform distribution on (a,b) has the density function

• See graphs of f(x) on page 264-265.
• Clearly, since f(x) is a constant, the area under f(x) over an interval (c,d) is just
• Easy to show using integration, that E(X) = (a+b)/2 and Var(X) = (b-a)²/12

The Normal Distribution

• We have already studied this. Nothing new.  Standard density function for the Standard Normal Distribution (Normal(0,1) -- i.e. with mean 0 and variance 1) is

• For Normal Distribution with mean m and variance s , we have

• We know that if X is Normal (m,s²), the standardized X* = (X-m)/s is Normal(0,1).

• In this example we are assuming that the mean is 1 Kilogram, and the SD is 20 micrograms.
• Problem 1:

• Problem 2: In 100 trial measurements what is the probability that more than 45 measurements will be within 10 micrograms? This is the Binomial(100, .3829) distribution, which can be approximated by the normal distribution with mean np and variance npq.
• Problem 3: What is the expected value (mean) of the absolute value of the size of the errors in a long series of measurements. Need to calculate E(|X|) where X = observed weight - 1 kilogram. X is Normal with mean 0 and SD 20. So we can write X = 20Z where Z is Normal(0,1).
• E(|X|)= 20E(|Z|) which is approximately 15.96 micrograms.

6. Hint: Want P(X<=0) = PX* <= (0-m)/s] = 1/3
and P(X <=1) = P(X* <= (1-m)/s] = 1. Use normal table to set up equations that you can solve for m and s\
7. Hint: Mean is 5'10". Use the fact that 10 percent of pop is over 6' tall and the Normal Table to calculate what the standard deviation must be. Next calculate the probability that an individual picked at random is over 6'2" using the Normal table. Finally the chance of 2 least successes in 100 trials of picking people at random is Binomial (n,p) where n = ____ and p = _____. This is well-approximated by the Normal distribution so again use the Normal table to calculate the answer
8.  Hint a) Just use Normal distribution table
b) You are finding P[ 11.8 < S₁₀₀/100 <12.2] which is Normal ( , ) (review pages 194-196)

Section 4.2 Exponential and Gamma Distributions

• Exponential( l ) Distribution: A random variable T (often associated with random time) has exponential distribution with rate l > 0 if T has probability density

• The exponential distribution models the waiting time until the next arrival of an event in a Poisson process. In such a process  the waiting times between each arrival are independent with the same exponential distribution.
• In such a Poisson process, with parameter l, the distribution of the number of arrivals in a fixed time interal of length t will be Poisson(lt), and the number of arrivals over disjoint time intervals will be independent.
• See page 289 for an excellent summary of properties of Poisson Arrival Process.
• Important Example 3 (page 284-285): Telephone calls at a switchboard arrive at an average rate of 3 per minute. N = Number of phone calls arriving in time period of length t minutes is Poisson(3t).
• W₁ = Waiting time in minutes for next phone call is Exponential(3).
• Note part f): P (Waiting time until fifth call is more than 2 minutes) = P(Number of calls that arrive in time interval of length 2 is less than or equal to 4)

• T_r ,  the waiting time for the arrival of the rth event after time 0 in a Poisson process with parameter l, has Gamma (r,l) distribution
• T_r = W₁ + W₂ + … + W_r, where each of the W_i are independent with exponential (l) distribution.

• The right tail probability is

• See page 286 for the mean and standard deviation. (Readily derived by integration )
• See page 287 for a graph of the Gamma density function for r = 1 to 10.
• Note since T_r is the sum of independent and identically distributed random variables as r gets larger the distribution approaches a Normal distribution by the Central Limit Theorem.
• The Gamma Distribution is also defined for non-integer values of r by a different formula.
(See page 291 top). We won't cover that.

Section 4.5 Cumulative Distribution Functions

• Cumulative Distribution Function (abbreviated c.d.f) is defined as
F(x) = P ( X<=x ).
• The c.d.f. F(x) uniquely determines the distribution of a random variable.
• This has meaning for both continuous and discrete random variables

For a  discrete random variable  X

• For discrete random variables, the c.d.f. is a step function that "jumps" at the points in the range of X  (values of X that occur with a non-zero probability).
• By the Fundamental Theorem of calculus,  for a continuous distribution, f(x) is the derivative of F(x) and F(x) an  antiderivative (indefinite integral ) of f(x) (The one satisfying F(-inf) = 0 and F(+inf) = 1.
• This means f(x) measures the rate of change of the c.d.f.  (See page 314)
• Example 1: The Uniform (0, 1) distribution

• See page 315 for sketch of density and c.d.f.

• For exponential(l) distribution we have already calculated the c.d.f F(x) =
1-e^-lx for x >= 0, 0 for x < 0.
• Example 2:  Uniform distribution on a disc:
• Pick a point at random on a unit disc.  calculate the c.d.f. and density function of X
• Let f(x) denote the density function.
• Then we know f(x)dx = P{X is in small interval dx} which is the shaded area/ area of unit circle.  The shaded area is 2*square-root(1-x²).  This leads to the density function f(x) as given.   The c.d.f. is found by integrating this (a nasty integral suitable  for calculating with the Ti-92 or Derive).

• Suppose X₁, …, X_n are independent with c.d.f.'s F₁(x), F₂(x), …, F_n(x)
• Distribution function of X_max, the maximum of X₁, …, X_n is
F_max(x) = P(X_max <= x) = P(X₁<= x, X₂ <= x, …, X_n <= x)
= P(X₁<= x) P(X₂ <= x) …P(X_n <= x) (by independence) =F₁(x)F₂(x)…F_n(x)
• Distribution function of X_min, the minimum of X₁, …, X_n is
F_min(x) = P(X_min <= x) = 1- P(X_min >= x) = 1 - P(X₁>= x, X₂ >= x, …, X_n >= x)
= 1 - P(X₁>= x) P(X₂ >= x) …P(X_n >= x) (by independence)
=1 - (1 - F₁(x))(1 - F₂(x))…(1 - F_n(x))
• Applying this to the exponential distribution, we get the minimum of independent exponential random variables is exponential with parameter l = l ₁ + l₂ + … + l _n (See example 3, page 317)
• Application to circuits: Example 4, page 317 - 318. Top and bottom loop each only last the minimum time of their respective components. Entire loop lasts as long as maximum of the two loops.
• Entire loop L has c.d.f function

• Percentiles and Distribution Functions: When we are asked to find the nth perctile for a particular distribution we mean are asked find x such that c.d.f F(x) = P(X<=x) is equal to n%. (i.e. for the 75^th percentile we need to find x such that F(x) = .75.
• We have already done problems like this for the Normal distribution (using normal table)and for the exponential distribution in section 4.2

• Exercises 1, 2, 3,  6, 8 Hints:
1.  We already did part a in class  -- just simple integration

2.  These are step function since discrete -- see examples in this section.
Sketch and also write down what F(x) is at each point that it jumps
For part b) note that the sum simplifies (see appendix page 516 bottom).

3 follows from example 2 in text a) is just a simple observation of the symmetry.
b)  Draw the sketch, then calculate the probability as a proportion of the unit circle area.

6a), b), and c)direct calculations  from definitions.d)  See page 316 -- X is the maximum {Y1, Y2, Y3}

8:  Just apply the ideas on maximum and minimum like in Example 4, page 317.
 

Radioactive Decay: