# 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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### 1 Answer

**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):}`