Download presentation

1
**Discrete Uniform Distribution**

The discrete uniform distribution occurs when there are a finite number (m) of equally likely outcomes possible. The pmf of a uniform discrete random variable X is: p(x) = 1 / m, where x=1,2,…,m The mean and variance of a discrete uniform random variable X are: µ = (m + 1) / 2 σ2 = (m2 - 1) / 12

2
**Bernoulli Distribution**

A random experiment with two possible outcomes that are mutually exclusive and exhaustive is called a Bernoulli Trial. One outcome is arbitrarily labeled a “success” and the other a “failure” p is the probability of a success q = 1 - p is the probability of a failure The Bernoulli random variable X assigns: X(failure) = 0 and X(success) = 1

3
**Bernoulli Distribution**

The pmf for a Bernoulli random variable X is: p(x) = px (1-p)1-x, where x=0,1 The mean and variance of a Bernoulli random variable X are: µ = p σ2 = pq

4
**Binomial Distribution**

A binomial experiment results from a sequence of n independent Bernoulli trials. The probability of success (p) remains constant in a binomial experiment The number of successes (X) is the random variable of interest in a binomial experiment If Y1, Y2, …, Yn are independent Bernoulli random variables, then X=∑ Yi is a binomial random variable.

5
**Binomial Distribution**

The pmf for a Binomial random variable X is: p(x) = nCx px (1-p)n-x, where x=0,1,…,n The mean and variance of a binomial random variable X are: µ = np σ2 = npq

6
**Hypergeometric Distribution**

The hypergeometric distribution applies when sampling without replacement from two possible mutually exclusive and exhaustive outcomes. Let X be the number of objects of type 1 drawn if n objects are drawn from N where there are M objects of type 1 and N-M objects of type 2. Then X is a hypergeometric random variable and the pmf of X is:

7
**Hypergeometric Distribution**

The mean and variance of a hypergeometric random variable X are: µ = n · (M/N) σ 2 = n · (M/N) · (1-M/N) · (N-n)/(N-1) As M and N converge to infinity and (M/N) converges to p, the hypergeometric distribution with n samples converges to the binomial distribution with n trials and p=M/N.

8
**Geometric Distribution**

A geometric distribution occurs when sampling independent Bernoulli trials. If X is the number of Bernoulli trials until the first success is observed, then X is a geometric random variable with pmf: p(x) = (1-p)x-1p, where x=1,2,3,… The mean and variance of a geometric random variable X are: µ = 1 / p σ2 = q / p2

9
**Geometric Distribution**

Notice that for integer k, P( X > k ) = qk P( X ≤ k ) = 1 – qk “Memoryless” or “No Memory” Property If X is a geometric random variable, then P( X > j + k | X > j ) = P( X > k ) This implies that in independent Bernoulli trails, there is no such thing as being “due” to observe a success.

10
**Negative Binomial Distribution**

A negative binomial distribution occurs when sampling independent Bernoulli trials. If X is the number of Bernoulli trials until the rth success is observed, then X is a negative binomial random variable with pmf: p(x) = x-1Cr-1 pr (1-p)x-r, where x=r,r+1,r+2,… The mean and variance of a negative binomial random variable X are: µ = r (1/p) σ2 = r (q / p2)

11
Poisson Distribution The Poisson distribution describes the number of occurrences of an event in a given time or on a given interval. Assumptions of a Poisson Process The number of events occurring in non-overlapping intervals is independent. The probability of 1 event occurring in a significantly short interval h is lh. The probability of 2 events occurring in a significantly short interval h is essentially zero.

12
**Poisson Distribution µ = s2 = l**

If X is defined to be the number of occurrences of an event in a given continuous interval and is associated with a Poisson process with parameter l>0, then X has a Poisson distribution with pdf: The mean and variance of a Poisson random variable X are: µ = s2 = l

13
Poisson Distribution If events of a Poisson process occur at a mean rate of l per unit, then the expected number of occurrences in an interval of length t is lt. Moreover, if Y is the number of occurrences in an interval of length t, it is Poisson with pdf:

14
Poisson Distribution The Poisson distribution with parameter l=np is useful for approximating the binomial distribution with sample size n and probability of success p in cases with sufficiently large sample size (n>20 and p<0.05). B(n,p) P(l=np) as n ∞, p , and np l

Similar presentations

OK

1 Engineering Statistics - IE 261 Chapter 3 Discrete Random Variables and Probability Distributions URL:

1 Engineering Statistics - IE 261 Chapter 3 Discrete Random Variables and Probability Distributions URL:

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Simple ppt on nuclear power plant Ppt on ministry of corporate affairs ontario Ppt on 9-11 conspiracy theories attacks videos Ppt on adr and drive download Ppt on idiopathic thrombocytopenia purpura splenectomy Ppt on review writing lesson Ppt on public speaking tips Ppt on carburetor working animation Ppt on marriott group of hotels Ppt on sea level rise florida