1. 5.2 Continuous Random Variable

2. Recall Discrete Distribution
For a discrete distribution, for example
Binomial distribution with n=5, and p=0.4, the probability distribution is x
f(x)

3. A probability histogramx
P(x)

4. How to describe the distribution of a continuous random variable?
For continuous random variable, we also represent probabilities by areas—not by areas of rectangles, but by areas under continuous curves.
For continuous random variables, the place of histograms will be taken by continuous curves.
Imagine a histogram with narrower and narrower classes. Then we can get a curve by joining the top of the rectangles. This continuous curve is called a probability density (or probability distribution).

5. Continuous distributions: Density Function
For any x, P(X=x)=0. (For a continuous distribution, the area under a point is 0.)
Can’t use P(X=x) to describe the probability distribution of X
Instead, consider P(a≤X≤b)

6. Density Function
A probability density function for a continuous random variable X is a nonnegative function f(x) with
And such that for all a≤b, one is willing to assign P[a≤X≤b] according to

7. Density function A curve f(x): f(x) ≥ 0 The area under the curve is 1
P(a≤X≤b) is the area between a and b

8. P(2≤X≤4)= P(2≤X<4)= P(2<X<4)

11. Cumulative Probability function
For X continuous with probability density f(x)
We can get the density function f(x) from F(x) by differentiation

16. The normal distribution
A normal curve: Bell shaped
Density is given by
μand σ2 are two parameters: mean and variance of a normal population
(σ is the standard deviation)

17. The normal—Bell shaped curve: μ=100, σ2=10

18. Normal curves: (μ=0, σ2=1) and (μ=5, σ 2=1)

19. Normal curves: (μ=0, σ2=1) and (μ=0, σ2=2)

20. Normal curves: (μ=0, σ2=1) and (μ=2, σ2=0.25)

21. The standard normal curve: μ=0, and σ2=1

25. Example (p323 #7)
In a grinding operation, there is an upper specification of 3.15 in. on a dimension of a certain part after grinding. Suppose that the standard deviation of this normally distributed dimension for parts of this type ground to any particular mean dimension μ is σ=.002 in. Suppose further that you desire to have no more than 3% of the parts fail to meet specifications. What is the maximum μ (minimum machining cost) that can be used if this 3% requirement is to be met?

27. So 3.15 is 1.88 σ above the mean.
*0.002=3.146

29. Exponential distribution
The exponential distribution is a continuous probability distribution with
Exponential distributions are often used to describe waiting times until occurrence of events.

30. Density curves

31. Mean and Variance Mean of an exponential distribution is
Variance of the exponential distribution is

32. Cumulative Probability Function

33. P(X<2) on density curve f(x) P(X<2) on CDF F(x)
When alpha=1
P(X<2) on density curve f(x)
P(X<2) on CDF F(x)

37. Force of Mortality Function
The force of Mortality Function is (p.760):
H(t)dt is the probability of dying in time t to t+dt if we are still living in t.
For exponential distribution
So the exponential distribution has a constant force-of-mortality.

38. If the lifetime of an engineering component is described using a constant force of mortality, there is no (mathematical) reason to replace such a component before it fails.
The distribution of its remaining life from any point in time is the same as the distribution of the time till failure of a new component of the same type.

40. Section 5.2.3 Weibull Distribution
Very commonly lifetimes of motors, etc. are modeled with Weibull distributions. A Weibull distribution is a generalization of an exponential distribution and provides more flexibility in terms of distributional shape.

41. For Weibull distribution

43. For component with increasing force-of-mortality (IFM) distribution, such components are retired from service once they reach a particular age, even if they have not failed.

44. Exercise
The lifetime of a certain type of battery has an exponential distribution with average lifetime 100 hours. 5 batteries are installed at the same time and suppose that the operations of the batteries are independent. Find the probability that only 2 batteries are still working after 50 hours.