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Uniform Distribution: Properties and Examples, Study notes of Physics

An introduction to the concept of uniform distribution, including its probability density function, cumulative distribution function, moments, and moment generating function. It also includes examples of uniform distributions and related problems. The document also mentions joint uniform distributions and provides examples of probability questions that can be solved using geometric methods.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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Uniform Distribution
f(x) =
1
baaxb
0 otherwise F(x) =
0xa
x
baa<xb
1x>b
E[X] = a+b
2Var(X) = (ba)2
12 MX(t) = etb eta
t(ba)*
Uniform distributions are those with constant density function over an interval. The constant is equal to
the reciprocal of the length of the integral, since the density function must integrate to 1.
Discrete distributions can also have uniform probability functions, such as the one that describes the prob-
ability of each outcome when rolling a standard die. These distributions are not discussed much in practice.
*Note: Moment Generating Functions will be discussed in a later section, but I have provided the MGF for
reference. Some people recommend you memorize all MGFs for common random variables. I would suggest
only memorizing a couple of them: the MGF for an exponential and for general discrete random variables.
As we progress through the course, I will let you know if there are others worth memorizing.
Joint Uniform Distribution: f(x, y ) = 1
[AREA OF RE GI ON ]
This is the easiest way to characterize joint uniform distributions. If asked for the probability of a subset,
B, of a uniform region, you can set up a double integral:
ZZ
B
1
T otal Area dA =1
T otal Area ZZ
B
dA =Area B
T otal Ar ea
Since the density is constant, we can pull it out of the integration, and we find the probability of a region is
just the ratio of its area to the area of the sample space. Because of this result, probability question with
joint uniform densities are often worked geometrically, rather than with integrals.
1
pf3
pf4
pf5

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Uniform Distribution

  • f (x) =

b − a

a ≤ x ≤ b

0 otherwise •^ F^ (x) =

0 x ≤ a x b − a

a < x ≤ b

1 x > b

• E[X] =

a + b 2 •^ Var(X) =

(b − a)^2 12

  • MX (t) =

etb^ − eta t(b − a)

Uniform distributions are those with constant density function over an interval. The constant is equal to the reciprocal of the length of the integral, since the density function must integrate to 1.

Discrete distributions can also have uniform probability functions, such as the one that describes the prob- ability of each outcome when rolling a standard die. These distributions are not discussed much in practice.

*Note: Moment Generating Functions will be discussed in a later section, but I have provided the MGF for reference. Some people recommend you memorize all MGFs for common random variables. I would suggest only memorizing a couple of them: the MGF for an exponential and for general discrete random variables. As we progress through the course, I will let you know if there are others worth memorizing.

Joint Uniform Distribution: f (x, y) =

[AREA OF REGION ]

This is the easiest way to characterize joint uniform distributions. If asked for the probability of a subset, B, of a uniform region, you can set up a double integral: ∫ ∫

B

T otal Area

dA =

T otal Area

B

dA =

Area B T otal Area

Since the density is constant, we can pull it out of the integration, and we find the probability of a region is just the ratio of its area to the area of the sample space. Because of this result, probability question with joint uniform densities are often worked geometrically, rather than with integrals.

Ex. Claim amounts are uniformly distributed along the interval [10, 60]. There is a policy limit on claims. What is the policy limit if the expected benefit is 31?

Ex. The losses on a policy are modeled by a uniform random variable on [0, 15000]. If there is a deductible of 1000, find the expected benefit payment.

Ex. Smith and Jones’ future lifetimes are independent and distributed uniformly between 5 to 20 years. What is the probability that they die within a year of each other?

Ex. Smith has 50,000 in a money market account, whose annual effective interest rate X is uniformly distributed on [0. 03 , 0 .05] and 30,000 in the stock market, whose annual effective interest rate Y is uniformly distributed on [− 0. 05 , 0 .1]. Smith knows that X and Y are independent. What is the probability that the interest accrued on his investment exceeds 3,000 in the next year?