Discrete Random Varibales

Question 1

WHAT is a random variable

Answer 1

a function that assigns a number to each outcome in the sample space

Question 2

a random varible can be discrete if it…….?

Answer 2

can only take on countable or finite number of possible outcomes

Question 3

A random variable is continuous if it……………?

Answer 3

has an uncountable possible outcomes ,and is usually generated by measuring

Question 4

A probability function must always obey the following two conditions

Answer 4

𝑓(𝑥)≥0 i.e. the probabilities are non negative for all 𝑥.

∑_(𝑖=1)^𝑛▒〖𝑃(𝑋=𝑥_𝑖 〗)=1 i.e. the probabilities add up to 1.

Question 5

what is a uniform descrete probability?

Answer 5

a probability distribution that has all outcomes equally likely.

Question 6

When calculating the mean from a probability distribution, it is usually referred to as the ?

Answer 6

expected Value

Question 7

When calculating the mean from a probability distribution, it is usually referred to as the

Answer 7

expected value

Question 8

The expected value is calculated by summing the products of each value of 𝑋 and its probability

Answer 8

E(𝑋) =∑_𝑥▒〖𝑥×P(𝑋=𝑥)〗
=∑_𝑥▒〖𝑥×𝑝(𝑥)〗

Question 9

Varience and standard deiation

Answer 9

these are measures of spread .
They are used to determine how close the mean is to the possible values of the random variable

Var(𝑋) =E[(𝑋−𝜇)^2 ]
=∑_𝑥▒〖(𝑥−𝜇)^

The variance is calculated by:
Var(𝑋) =E[(𝑋−𝜇)^2 ]
=∑_𝑥▒〖(𝑥−𝜇)^2×𝑝(𝑥)〗

Question 10

Effects of a linear change on a discrete random variable

Answer 10

A random variable 𝑋 has a probability distribution of P(𝑋=𝑥) has and expected value of E(𝑋), standard deviation of SD(𝑋). and variance of Var(𝑋).
If a random variable 𝑌 has a probability distribution of P(𝑌=𝑎𝑥+𝑏) the expected value of E(𝑌), standard deviation of SD(𝑌) and variance of Var(𝑌) are as follows.
**E(𝑌)=𝑎E(𝑋)+𝑏
SD(𝑌)= 𝑎SD(𝑋)
Var(𝑌)=𝑎^2 Var(𝑋) **

Question 11

Bernoulli sequence

Answer 11

A Bernoulli sequence is the name used to describe a sequence of repeated trials with the following properties:
* Each trial results in one of two outcomes, which are usually designated as either a success, S , or a failure, F.
* The probability of success on a single trial, p, is constant for all trials (and thus the probability of failure on a single trial is 1−𝑝).
* The trials are **independent (so that the outcome of any trial is not affected by the outcome of any previous trial).
**

Question 12

summary for bernoulli random variables

Answer 12

The long-term mean, or expected value 𝐸(𝑋), of a Bernoulli distribution with parameter 𝑝 is 𝑝.(which is the success probability)
The variance of a Bernoulli distribution with parameter** 𝑝 is 𝑝(1 − 𝑝).**
A Bernoulli random variable can be used as the probability model for situations involving two mutually exclusive outcomes.

Question 13

The binomial probability distribution

Answer 13

The number of successes in a Bernoulli sequence of 𝑛 trials is called a binomial random variable and is said to have a binomial probability distribution

Question 14

The binomial probability distribution

Answer 14

The number of successes in a Bernoulli sequence of 𝑛 trials is called a binomial random variable and is said to have a binomial probability distribution.
So
* There are a fixed number of trials (a fixed sample size).

• On each trial, the event of interest either occurs or does not occur (success or failure) and the probability of occurrence (or not) is the same on each trial.
  * Trials are independent of one another.