Common Univariate

Question 1

How to denote prob. of success

Answer 1

P, with value 1

Question 1

how many outcomes does bernuolli rv has and name them?

Answer 1

2 outcomes, success or failure

Question 2

how to denoted prob. of failure in bernoulli rv

Answer 2

1-P , with value 0

Question 3

bernuolli is used for which type of variable?

Answer 3

discrete

Question 4

Probability Mass Function (PMF)
For a Bernoulli random variable
𝑋

Answer 4

P(X=1)=p
𝑃(𝑋=0)=1−𝑝

PMF is given by:
f(x)=p^x* (1−p)^1−x

This means:

When x=1: 𝑓(1)=𝑝
x=0: 𝑓(0)=1−𝑝
^= power
*= multiplication

Question 5

Mean (Expected Value) of bernuolli rv

Answer 5

μx=p

Question 6

variance of bernuolli rv

Answer 6

Var(X)=p(1−p)

Question 7

CDF of a Bernoulli random variable
𝑋
X is a step function that increases from 0 to 1 at
x=0:

Answer 7

For
𝑥<0: 𝐹(𝑥)=0
For
0≤x<1: F(x)=1−p
For
x≥1: F(x)=1

Question 8

The CDF is a piecewise function:

Answer 8

Before
x=0, the probability is 0.
Between
x=0 and x=1, the probability jumps to
1−p.
At x=1 and beyond, the probability is 1.

Question 9

the variance in bernuolli rv

Answer 9

the variance is low for values of p close to 1 or 0, and the maximum variance is at 0.5

Question 10

example for bernuolli rv

Answer 10

Let’s say we are modeling whether a firm will default on its debt over a year:

Let
X=1 represent the firm defaulting (success).
Let
X=0 represent the firm not defaulting (failure).
Suppose the probability of default (p) is 0.2.
For this example:

P(X=1)=p=0.2
P(X=0)=1−p=0.8
The PMF would be:
For
X=1:
f(1)=0.2
For
X=0:
f(0)=0.8

Mean (Expected Value) and Variance

The mean (expected value)= UX=P
For our example,
𝜇𝑋=0.2

The variance of
X is: Var(X)=p(1−p)
For our example,
Var(X)=0.2×0.8=0.16.

CDF:
For our example:

For 𝑥<0:F(x)=0
For 0≤x<1: F(x)=0.8
For x≥1: F(x)=1

Question 11

the probability mass function (PMF) for this Bernoulli variable is
defined only for which value

Answer 11

for 0 or 1

Question 12

cdf for bernoulli defined for which no.

Answer 12

for all real no.

Question 13

binomial probability function defines

Answer 13

the probability of exactly x
successes in n bernoulli trials with probabiltiy p constant

Question 14

binomial probabiltiy function

Answer 14

P(X=x)= (
x
n)
​p ^x (1−p) ^n−x

(n
x)= ncx= n!/(n-x)! * x!

Question 15

expected value of x or expected value of success in binomial

Answer 15

np

Question 16

variance in binomial

Answer 16

var(X)= np (1-p)

Question 17

what expected value help and what is expected value is means in binomial

Answer 17

expected value is mean here. also it helps to set epextation for the outcome

Question 18

what variance in binomial helps to define

Answer 18

fluctuation

Question 19

why we do ncx in binomial rv?

Answer 19

binomial coefficient is essential for determining the number of ways to arrange
x successes in
n trials, which is a crucial component of the binomial probability formula.

Question 20

Binomial distributions are used extensively in ?

Answer 20

the investment world where outcomes
are typically seen as successes or failures. In general, if the price of a security goes up, it
is viewed as a success. If the price of a security goes down, it is a failure. In this context,
binomial distributions are often used to create models to aid in the process of asset
valuation

Question 21

The Poisson distribution is which one discrete or continuous?

Answer 21

discrete probability distribution with a number of real world applications

Question 22

An interesting feature of the Poisson distribution is that both its mean and variance are
equal to ?

Answer 22

lambda (λ)

Question 23

Define poisson distribution

Answer 23

The Poisson distribution is a discrete probability distribution that describes the likelihood of a given number of events happening within a fixed interval of time or space

Question 24

what the key parameter of the Poisson distribution λ (lambda) represents?

Answer 24

the average rate of occurrence of the events

Question 25

Poisson Distribution Formula

Answer 25

P(X=k)= λ ^k *e ^−λ/ k! in calculator , ​e^-lambda = digit,+-, 2nd e^x k is the number of events. λ is the average rate (mean) of events per interval.

Question 26

Mean and Variance of poisson distribution

Answer 26

Mean (𝜇) = λ Variance (𝜎2)= λ

Question 27

what is continuous uniform distribution?

Answer 27

probability distribution in which all outcomes within a specific range are equally likely. This range is defined by two parameters: the lower limit a and the upper limit b.

Question 28

Properties of the Continuous Uniform Distribution

Answer 28

Range: The distribution is defined over the interval [a,b], meaning outcomes can only occur within this range. Probability of Specific Outcomes: For any specific value x where ab)=0 The probability that X falls outside the interval [a,b] is zero. Probability within an Interval: For any a≤x1

Question 29

Probability Density Function (PDF) for conti. uniform rv?

Answer 29

f(x)={1/ b−a} for a≤x≤b 0 otherwise, This indicates that the density is constant across the interval [a,b] and zero outside of it. ​

Question 30

Cumulative Distribution Function (CDF) for continuous uniform distribution

Answer 30

F(x)= 0 for xb ​ This function is linear between a and b and reaches 0 and 1 at the boundaries a and b, respectively.

Question 31

mean of conti. uniform distribution

Answer 31

a+b/2

Question 32

variance of conti. uniform

Answer 32

(b-a)^2/ 12

Question 33

about conti uniform distribution

Answer 33

The simplicity of the uniform distribution lies in its equal probability for all outcomes within the range [a,b], making it straightforward to understand and use in practical situations. The CDF is a straight line, and the PDF is a constant value over the interval. The mean is the midpoint of the interval, and the variance measures the spread of the distribution.