statistics - topic 3 -probabilty distributions

Question 1

what is a random variable?

Answer 1

Random variables – represents a possible numerical value from a random experiment

Question 2

what is a discrete random variable?

Answer 2

Discrete random variable takes on on more than a countable number of values

Question 2

what is a discrete random variable?

Answer 2

Discrete random variable takes on on more than a countable number of values

Question 3

what is the probability distribution function?

Answer 3

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

Question 4

what are the properties of a probability distribution function?

Answer 4

0≤P(x)≤1 for any value of x

The individual probabilities sum to 1: ∑ P(x) =1

Question 5

What is The cumulative probability?

Answer 5

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

Question 6

What is the mean of a discrete random variable:

Answer 6

E[X]=∑ [xP(x)]

Question 7

What is the variance of a discrete random variable X:

Answer 7

σ^2=E[(X-μ )^2 ]=∑〖(x-μ )^2 P(x) 〗

Question 8

What is the standard deviation of a discrete random variable X?

Answer 8

σ=√(σ^2 )=√(∑〖(x-μ )^2 P(x) 〗)

Question 9

How would you find the expected value of a function g(x) with a probalility function P(x)?

Answer 9

E[g(x)= ∑g(x)P(x)

Question 10

What is the covariance between X and Y

Answer 10

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

Question 11

What is the correlation between X and Y:

Answer 11

𝜌= 𝐶𝑜𝑟𝑟(𝑋,𝑌) =𝐶𝑜𝑣(𝑋,𝑌)/(𝜎_𝑋 𝜎_𝑌 )

Question 12

what is the covariance and correlation when two random variables are statiscally independent?

Answer 12

If two random varibles are statistically independent then the covariance and correlation between them is 0

Question 13

what are the two forms of probability distributions?

Answer 13

there is discrete probability distributions and there are continous probability distributions

Question 14

give an example of a discrete probability distribution?

Answer 14

binomial distribution

Question 15

give an example of a continous probability distribution?

Answer 15

uniform and normal distributions

Question 16

what is a bernoulli distribution?

Answer 16

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?

Question 17

what is the mean of a Bernoulli probability distribution ?

Answer 17

p

Question 18

what is the variance of a Bernoulli probability distribution

Answer 18

p(1-p)

Question 19

what are the features of a binomial distribution?

Answer 19

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

Question 20

what is the probability of x successes in n trials in a binomial distribution?

Answer 20

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’

Question 21

what is the mean of a binomially distributed variable ?

Answer 21

mean is equal to np

Question 22

what is the standard deviation of a binomially distributed variable?

Answer 22

√(𝑛𝑝(1−𝑝) )

Question 23

what is a continous random variable?

Answer 23

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

Question 24

what are the properties of the probability density function?

Answer 24

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

Question 25

what is the area under the curve between a and b of a probabilty distribution function?

Answer 25

the shaded area under the curve is the probability that X is between a and b

Question 26

what is the mean in a continous random variable?

Answer 26

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

Question 27

what is the varience of X in a continous random variable?

Answer 27

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〗

Question 28

what is the covariance between two continuous random variables X and Y?

Answer 28

The covariance between 𝑋 and 𝑌 is:
𝐶𝑜𝑣(𝑋,𝑌)=𝐸[(𝑋−𝜇_𝑋 )(𝑌−𝜇_𝑌 )]

Question 29

what is the correlation between two continuous random variables X and Y?

Answer 29

The correlation between 𝑋 and 𝑌 is:
𝜌=𝐶𝑜𝑟𝑟(𝑋,𝑌)=𝐶𝑜𝑣(𝑋,𝑌)/(𝜎_𝑋 𝜎_𝑌 )

Question 30

if the continous random variables X and Y are independent, then what is the covariance?

Answer 30

If the random variables 𝑋 and 𝑌 are independent, then the covariance between them is 0

Question 31

what is a uniform distribution ?

Answer 31

The uniform distribution is a probability distribution that has equal probabilities for all equal-width intervals within the range of the random variable, 𝑋

Question 32

what can a density function of a uniform distribution be represented as

Answer 32

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

Question 33

what is the mean of a uniformly distributed variable?

Answer 33

𝜇=(𝑎+𝑏)/2

Question 34

what is the variance of a uniformly distributed variables?

Answer 34

𝜎^2=(𝑏−𝑎)^2/12

Question 35

what are the properties of the normal distribution?

Answer 35

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 

Question 36

how do you define the normal distribution?

Answer 36

𝑋~𝑁(𝜇,𝜎^2 )

Question 37

what is the formula for the normal probability density function?

Answer 37

𝑓(𝑥)=1/√(2𝜋𝜎^2 ) 𝑒^(〖−(𝑥−𝜇)^2〗∕〖2𝜎^2 〗)

Question 38

what is a standardised normal distribution?

Answer 38

the standardized normal distribution (𝑍), with mean 0 and variance 1, which we denote by 𝑍~𝑁(0,1)

Question 39

how can you transfer the X into a standard normal distribution

Answer 39

We transform 𝑋 units into 𝑍 units by subtracting the mean of 𝑋 and dividing by its standard deviation
𝑍=(𝑋−𝜇)/𝜎

Question 40

what is the formulae to find the area between a and b ?

Answer 40

𝑃(𝑎<𝑋<𝑏)=𝑃((𝑎−𝜇)/𝜎<𝑍<(𝑏−𝜇)/𝜎)

Question 41

what are the steps to find the X value for a known probability?

Answer 41

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

Question 42

what is the formula for the varience of two random variables (𝜎_𝑊^2)?

Answer 42

𝜎_𝑊^2=𝑎^2 𝜎_𝐴^2+𝑏^2 𝜎_𝐵^2+2𝑎𝑏𝐶𝑜𝑟𝑟(𝐴,𝐵) 𝜎_𝐴 𝜎_𝐵

Question 43

how can you convert the binomial distribution into a normal distribution?

Answer 43

𝜇=𝑛𝑝
𝜎^2=𝑛𝑝(1−𝑝)

Question 44

when is the shape of a binomial distribution approxiametly normal?

Answer 44

The shape of the binomial distribution is approximately normal if 𝑛𝑝>5 and n(1−𝑝)>5