Let X be a random variable Write down for each of the following distibution of X whether its a discrete or a continuous distribution:  X is uniform, Possion, normal, Laplace, Bernoulli, Binomial,...

Let X be a random variable

Write down for each of the following distibution of X whether its a discrete or a continuous distribution:

 

X is uniform, Possion, normal, Laplace, Bernoulli, Binomial, geometric, hypergeometric, or exponential-distribution

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tiburtius | High School Teacher | (Level 2) Educator

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Uniform random variable can be both continuous and discrete. In discrete case the probability of each of `n` events is `1/n` and cumulative distribution function is

 `F(x;a,b)={(0 ,if x<a),((|__x__|-a+1)/(b-a+1) ,if a leq x <b),(1 ,if x geq b) :}`

In continuous case cumulative distribution function is

 `F(x)={(0, if x<a),((x-a)/(b-a), if a leq x<b),(1, if x geq b):}`

Poisson random variable is discrete random variable with probability density function

`f(k;lambda)=(lambda^ke^(-lambda))/(k!)` `,lambda>0`

Normal random variable (Gaussian distribution) is continuous random variable which you can see from its probability density function

`f(x)=1/(sigma sqrt(2pi))e^(-((x-mu)^2)/(2sigma^2))`

where `mu` is mean and `sigma^2` is variance.

Laplace random variable (double exponential) is continuous random variable with probability density function

`f(x;mu,b)=1/(2b) e^(-(|x-mu|)/b)`

where `mu` is mean and variance is `2b^2`.

Bernoulli random variable is probably the simplest discrete random variable. It can take one of only two values ( like coin toss - heads or tails) 1 - success and 0 - failure. Probability density function:

`f(k;p)={(p ,if k=1),(1-p ,if k=0):}` where `p` is probability of success.

Binomial random variable is random variable of `n` independent Bernoulli trials (`n` coin tosses) and is, as such, discrete. Probability density function:

`f(k;n,p)=((n),(k))p^k(1-p)^(n-k)`

Geometric random variable models the number of fails before first success of repeated Bernoulli trials and is, as such, discrete. Probability density function:

`f(k;p)=(1-p)^(k-1)p`  

Hypergeometric is like geometric random variable discrete with probability density function

`f(k;N,n,K)=(((K),(k))((N-K),(n-k)))/(((N),(n)))`

where `N` is population size, `K` number of success states in population, `n` number of draws and `k` is of course number of successes.

Exponential random variable is (similar to Laplace) continuous with probability density function

`f(x;lambda)={(0, if x<0),(lambda e^(-lambda x), if x geq 0):}`

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