Data Science

Question 1

How is the mean and variance of two 🎃INDEPENDENT, NORMALLY DISTRIBUTED🎃 variables calculated? calculate for both addition and subtraction

Answer 1

X+Y=T mean(T)= mean(X)+mean(Y) var(T)^2=var(X)^2+var(Y)^2

X-Y=Z mean(Z)= mean(X)-mean(Y) var(Z)^2=var(X)2+var(Y)^2

Question 2

Write the formula of mean and variance, using expected value:

Answer 2

Expected value is basically the same as mean, now:
Here’s how it’s calculated:
mean=E(x) = x*p(x) for all values of x (if they are discrete we use sigma, if they are continuous, we use integral)

For variance: variance =E ((x-E(x))^ 2) (mean of this variable: (x- mean (x))^2 )

Question 3

What does central limit theorem say?

Answer 3

The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger,
🧨 regardless of the population’s distribution. 🧨
Sample sizes equal to or greater than 30 are often considered sufficient for the CLT to hold.

Question 4

What is a latent variable?

Answer 4

In statistics, latent variables are variables that are not directly observed but are rather inferred through a mathematical model from other variables that are observed

Question 5

What do we mean when we say the sample mean targets the population mean?

Answer 5

Means that it’s a good estimate of the population mean

Question 6

What is the variance of the sampling distribution of means telling us? (Considering CLT)

Answer 6

The formula for variance of the sampling distribution of means is: population variance/n, so the larger the sample size, the smaller the variance and the closer the means of samples to the population mean. In extreme form, n is the whole population, and whatever number of samples we take, we’ll have one mean only (variance really small).

Question 7

How the sampling distribution would look, if the original distribution is not normal?

Answer 7

It approaches a normal distribution.

Question 8

What is the relation between non-normal original distribution and the suitable sample size?

Answer 8

The further the original distribution is from normal, the larger the sample size should be so that the sampling distribution of means approaches normal distribution (typically sample size>=30 will approach normal distribution)

Question 9

If the original distribution is normal, the sampling distribution of the means will also be normal. True or False?

Answer 9

True

Question 10

Where do we use CLT?

Answer 10

When sample is not normally distributed, we can still use some tools designed for normal distribution using CLT.

Question 11

Summarize the 3 parts of CLT; which parts hold true for any sample size?

Answer 11

1-The mean of the sampling distribution of means is a fair estimate of the mean of the population from which the samples were drawn

2-The variance of the sampling distribution of means is a fair estimate to the variance of the population from which the samples were drawn, divided by n

3-If the original distribution is normal, the sampling distribution of the means will also be normal. Otherwise, if n>=30, we can still safely assume normal.

Part 1 and 2, are true for any sample size

Question 12

What is the sampling distribution of the mean?

Answer 12

The distribution of the means of the samples taken from the original data. (each sample has a mean, sampling distribution of the means, is the distribution patterns of the means of all the samples taken)

Question 13

What is the formula for calculating variance for continuous and discrete variables?

Answer 13

f(x) is the probability distribution function
p(x) is the probability of each discrete value the variable can take
🤓
Continuous Variables:
∫ (x-mean)^2*f(x) dx
🤓
Discrete Variables:
∑ (xi-mean)^2*pi(xi)

Question 14

How does adding/subtracting a constant to/from a variable change its variance?

Answer 14

It doesn’t change its variance

Question 15

How does multiplying a variable by a constant change its variance?

Answer 15

The new variance= the old variance* constant^ 2

Question 16

How does the variance change when we have a set of INDEPENDENT variables added together?

Answer 16

When they are INDEPENDENT we add the variance of each variable:
var (x±y) =var(x) + var(y)

Question 17

How does the variance change when we have a set of DEPENDENT variables added/subtracted?

Answer 17

var (x+y) =var(x) + var(y) + 2 cov(x, y) (cov=covariance)

var (x-y) =var(x) + var(y) - 2 cov(x, y) (cov=covariance)

Question 18

How is the STD of x+y calculated?

Answer 18

1-Calculate x+y variance: var(x) + var(y)

2- Take the square root of the x+y variance: √ (var(x) + var(y))

Question 19

What is The Most Important Probability Distribution for Discrete Random Variables?

Answer 19

When a random variable follows a binomial distribution

Question 20

What conditions should be met before being certain a distribution is binomial?

Answer 20

1- Probability of success in each separate trial is the same
2- Trials are independent: the result of one trial doesn’t depend on others
3- Fixed number of trials
4- Each trial can be classified as either fail or success

Question 21

What is the formula for getting x number of successes with n trials in a binomial distribution? And the shorthand of it?

Answer 21

P(x) = [n!/x!(n-x)!] p^{x} q^{n-x}
X ~ B(N,P)
X is a binomial random variable with N trials and success probability of P

Question 22

What’s the equation for the cumulative probability distribution for binomial distributions?

Answer 22

P(X<=x) =∑ [n!/k!(n-k)!] p^{k} q^{n-k} 0

Question 23

On which parameter does the shape of Binomial probability distribution (probability vs number of trials) depend? How does it change in relation to the change of this parameter?

Answer 23

P, the probability of success the higher the P(closer to 1), more left skewed the probability distribution is. When P is around .5, it’s almost symmertic.
Explanation: If P is close to one, then let’s say we have 10 trials and we start by 1, meaning that: the probability of just having 1 success in 10 trials, it’s going to be really small since there’s a high chance we succeed more than 1 time. So it’s going to get bigger as we increase the number of successes. Therefore it’s going to be left skewed.

Question 24

Binomial probability distribution is discrete. True or False?

Answer 24

True

Question 24

Explain how the binomial probability distribution is plotted

Answer 24

binomial distribution is made of p( number of successes) on y axis and number of trials on the x axis. for example if the x axis has 20 ticks, it means we have 20 trials, if we want to plot the probability of having one success among 20 trials, it would be y=p(X=1) , x=1

Question 25

How is the mean and std of a binomial variable is calculated? What do they depend on?

Answer 25

They depend on the probability of success
n= number of trials
p= probability of success
mean=n*p
std=√(np(1-p))

Question 26

What is the 10% rule for assuming independence between trials?

Answer 26

We can make inferences based on things being close to a binomial distribution or a normal distribution. In case of binomial distribution, if the sample is less than or equal to 10% of the population, then it’s ok to assume approximate independence.

Question 27

What does binompdf and binomcdf functions take as input and what’s their output?

Answer 27

Binompdf and Binomcdf: both take n: number of trials, p: probability of success, x: number of success.
Binompdf output: the probability of exactly x times of success happening out of n trials
Binomcdf output: the cumulative probability of up to and including x times of success happening out of n trials.

Question 28

What is Bernoulli distribution?

Answer 28

The Bernoulli distribution describes events having exactly two outcomes, which are ubiquitous in real life. Some examples of such events are as follows: a team will win a championship or not, a student will pass or fail an exam, and a rolled dice will either show a 6 or any other number.