statistics - topic 3 -probabilty distributions Flashcards
what is a random variable?
Random variables – represents a possible numerical value from a random experiment
what is a discrete random variable?
Discrete random variable takes on on more than a countable number of values
what is a discrete random variable?
Discrete random variable takes on on more than a countable number of values
what is the probability distribution function?
The probability distribution function, P(x) , of a discrete random variable X represents the probability that X takes the value x , as a function of x . That is, P(x) = P (X = x) , for all values of x
what are the properties of a probability distribution function?
0≤P(x)≤1 for any value of x
The individual probabilities sum to 1: ∑ P(x) =1
What is The cumulative probability?
The cumulative probability function, denoted F(x), shows the probability that X does not exceed the value x, F(x)=P(X≤x), where the function is evaluated at all values of x
What is the mean of a discrete random variable:
E[X]=∑ [xP(x)]
What is the variance of a discrete random variable X:
σ^2=E[(X-μ )^2 ]=∑〖(x-μ )^2 P(x) 〗
What is the standard deviation of a discrete random variable X?
σ=√(σ^2 )=√(∑〖(x-μ )^2 P(x) 〗)
How would you find the expected value of a function g(x) with a probalility function P(x)?
E[g(x)= ∑g(x)P(x)
What is the covariance between X and Y
Cov(X,Y)=E[(X-μ_X )(Y-μ_Y )]=∑x ∑y (x-μ_X )(y-μ_Y )P(x,y) where P(x,y) denotes the probability that X=x and Y=y
What is the correlation between X and Y:
𝜌= 𝐶𝑜𝑟𝑟(𝑋,𝑌) =𝐶𝑜𝑣(𝑋,𝑌)/(𝜎_𝑋 𝜎_𝑌 )
what is the covariance and correlation when two random variables are statiscally independent?
If two random varibles are statistically independent then the covariance and correlation between them is 0
what are the two forms of probability distributions?
there is discrete probability distributions and there are continous probability distributions
give an example of a discrete probability distribution?
binomial distribution
give an example of a continous probability distribution?
uniform and normal distributions
what is a bernoulli distribution?
it is a special case of the binomial distribution is one. it has two outcomes sucess or failure. the probability of sucess is equal to p and failure is equal to p-1. X is defined as x=1 if it is sucess or x=0 if it is failure?
what is the mean of a Bernoulli probability distribution ?
p
what is the variance of a Bernoulli probability distribution
p(1-p)
what are the features of a binomial distribution?
a fixed number of trials n
two mutually exclusive and collectively exhaustive categories
probability of sucess is p and failure 1-p
constant probability for each trial
the trials are independent
what is the probability of x successes in n trials in a binomial distribution?
𝑃(𝑘)= 𝑛! / [𝑘!(𝑛−𝑘)!] x 𝑝^𝑘 (1−𝑝)^(𝑛−𝑘)=𝐶_𝑘^𝑛 𝑝^𝑘 (1−𝑝)^(𝑛−𝑘)
𝑝 = the probability of success on each trial
𝑘 = number of ‘successes’
𝑛 = sample size (number of trials or observations)
𝑃 = probability of ‘success’
what is the mean of a binomially distributed variable ?
mean is equal to np
what is the standard deviation of a binomially distributed variable?
√(𝑛𝑝(1−𝑝) )
what is a continous random variable?
a continous random variable is a variable that can take any value in an interval ie the size of an item or the time required to produce a unit of output
what are the properties of the probability density function?
𝑓(𝑥)>0 for all values of 𝑥
The area under the probability density function 𝑓(𝑥) over all values of 𝑋 equals 1
The probability that 𝑋 lies between two values is the area under the density function between the two values:
𝑃(𝑎<𝑋<𝑏)=∫_𝑎^𝑏▒𝑓(𝑥) 𝑑𝑥
The cumulative density function 𝐹(𝑥) is the area under the probability density function 𝑓(𝑥) from the minimum value, 𝑥_𝑚𝑖𝑛, up to 𝑥:
𝐹(𝑥)=∫_(𝑥_𝑚𝑖𝑛)^𝑥▒𝑓(𝑥) 𝑑𝑥
what is the area under the curve between a and b of a probabilty distribution function?
the shaded area under the curve is the probability that X is between a and b
what is the mean in a continous random variable?
The mean of 𝑋, denoted 𝜇_𝑋, is defined as the expected value of 𝑋:
𝜇_𝑋=𝐸[𝑋]=∫_(𝑥_𝑚𝑖𝑛)^(𝑥_𝑚𝑎𝑥)▒𝑥𝑓(𝑥)𝑑𝑥
what is the varience of X in a continous random variable?
The variance of 𝑋, denoted 𝜎_𝑋^2, is defined as the expectation of the squared deviation, (𝑋−𝜇_𝑋 )^2, of a random variable from its mean:
𝜎_𝑋^2=𝐸[(𝑋−𝜇_𝑋 )^2 ]=∫_(𝑥_𝑚𝑖𝑛)^(𝑥_𝑚𝑎𝑥)〖(𝑥−𝜇_𝑥 )^2 𝑓(𝑥)𝑑𝑥〗
what is the covariance between two continuous random variables X and Y?
The covariance between 𝑋 and 𝑌 is:
𝐶𝑜𝑣(𝑋,𝑌)=𝐸[(𝑋−𝜇_𝑋 )(𝑌−𝜇_𝑌 )]
what is the correlation between two continuous random variables X and Y?
The correlation between 𝑋 and 𝑌 is:
𝜌=𝐶𝑜𝑟𝑟(𝑋,𝑌)=𝐶𝑜𝑣(𝑋,𝑌)/(𝜎_𝑋 𝜎_𝑌 )
if the continous random variables X and Y are independent, then what is the covariance?
If the random variables 𝑋 and 𝑌 are independent, then the covariance between them is 0
what is a uniform distribution ?
The uniform distribution is a probability distribution that has equal probabilities for all equal-width intervals within the range of the random variable, 𝑋
what can a density function of a uniform distribution be represented as
The density function of a variable that has a uniform distribution can be represented as:
𝑓(𝑥)={█(1/(𝑏−𝑎) if 𝑎≤𝑥≤𝑏@0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒) ┤
where 𝑎 is the minimum value and 𝑏 is the maximum value of 𝑋
what is the mean of a uniformly distributed variable?
𝜇=(𝑎+𝑏)/2
what is the variance of a uniformly distributed variables?
𝜎^2=(𝑏−𝑎)^2/12
what are the properties of the normal distribution?
The normal distribution is:
‘Bell Shaped’
Symmetric
The area under the curve is 1 so half is above the mean, half is below
Location is determined by the mean, 𝜇
Spread is determined by the standard deviation, 𝜎
The random variable has an infinite theoretical range: + to
how do you define the normal distribution?
𝑋~𝑁(𝜇,𝜎^2 )
what is the formula for the normal probability density function?
𝑓(𝑥)=1/√(2𝜋𝜎^2 ) 𝑒^(〖−(𝑥−𝜇)^2〗∕〖2𝜎^2 〗)
what is a standardised normal distribution?
the standardized normal distribution (𝑍), with mean 0 and variance 1, which we denote by 𝑍~𝑁(0,1)
how can you transfer the X into a standard normal distribution
We transform 𝑋 units into 𝑍 units by subtracting the mean of 𝑋 and dividing by its standard deviation
𝑍=(𝑋−𝜇)/𝜎
what is the formulae to find the area between a and b ?
𝑃(𝑎<𝑋<𝑏)=𝑃((𝑎−𝜇)/𝜎<𝑍<(𝑏−𝜇)/𝜎)
what are the steps to find the X value for a known probability?
Steps to find the 𝑋 value for a known probability:
1. Find the 𝑍 value for the known probability
2. Convert to 𝑋 units using the formula: 𝑋=𝜇+𝑍𝜎
what is the formula for the varience of two random variables (𝜎_𝑊^2)?
𝜎_𝑊^2=𝑎^2 𝜎_𝐴^2+𝑏^2 𝜎_𝐵^2+2𝑎𝑏𝐶𝑜𝑟𝑟(𝐴,𝐵) 𝜎_𝐴 𝜎_𝐵
how can you convert the binomial distribution into a normal distribution?
𝜇=𝑛𝑝
𝜎^2=𝑛𝑝(1−𝑝)
when is the shape of a binomial distribution approxiametly normal?
The shape of the binomial distribution is approximately normal if 𝑛𝑝>5 and n(1−𝑝)>5
