Random variable

Question 1

PMF Talks about

Answer 1

Mass, height, for discrete random variable, gives probability at a point

Question 2

PMF FUNCTION

Answer 2

F(X)= p(X=x) , X= discrete rv, x=event, F(x)= function

Question 3

PMF sum of probabilty

Answer 3

1

Question 4

properties of PMF

Answer 4

1. P(X is greater than equal to 0)
2. sum of P(x) = 1

Question 5

CDF

Answer 5

gives probablity that random variable is less than or equal to x

Question 6

CDF f(x)?

Answer 6

f(x) = P(X is less than equals to x)

Question 7

for discrete random variable cdf is?

Answer 7

step up function because it increases as the value of x increases

Question 8

Properties of CDF

Answer 8

1. Non decreasing because of step up function where f(x1) less than equals to f(x2) less than equals to f(x3) if x1<x2,x3
2. x= -ve infinity f(x)=0
   x= +ve infintiy f(x)=1

Question 9

PDF

Answer 9

Calculate the prob. of an outcome over 2 interval

Question 10

Where prob. lies in PDF?

Answer 10

Under the area of curve

Question 11

where the prob. lies in pmf?

Answer 11

at the height of interval

Question 12

pdf is calculated for?

Answer 12

continuous rv

Question 13

continuous rv takes on value withing?

Answer 13

range

Question 14

pdf f(x) is greater than equals to 1 , where f(x) can be between +ve infinity to -ve infinity

Answer 14

in this situation f(x)dx =1 where the area under the curve over all possible value of x=1

Question 15

pdf what height or density

Answer 15

density, depthness matters

Question 16

discrete rv

Answer 16

countable no. of possible outcome

Question 17

how many values can discrete ev take?

Answer 17

2 values, its 0 & 1

Question 18

discrete rv also refered as

Answer 18

bernouli rv

Question 19

continuous rv

Answer 19

uncountable no./ infinite no. of possible outcomes

Question 20

prob. for particualar no. for continuous rv is

Answer 20

0 , because there are infinite no. of possible outcome and value for 1 outcome is impossible to determine

Question 21

continuos rv measure prob over

Answer 21

+ve interval

Question 22

what is expectation means?

Answer 22

the expected value is weighted average of possible outcome means here we give weights to some x random variable. weight= prob. that outcome will occur

Question 23

formula for e(X)

Answer 23

sumP(Xi)Xi= P(X1)x1+P(X2)x2+…..+P(Xn)Xn

Question 24

properties of e(x)

Answer 24

1. e(x+y)= e(x)+ e(y)
2. e(cX)= c*E(X)

Question 25

Linear transformation

Answer 25

when we know the linear relationship between x &y and we know the mean, sd, variance, skewness and kurtosis for x we don't need to calculate same for y differently because there are formula or equation for each to find out the value for y

Question 26

variance

Answer 26

Y = b square *variance of x

Question 27

mean of y

Answer 27

E(Y) = a + bE(X)

Question 28

skewness b>0

Answer 28

skew of y = skew of x

Question 29

skewness b<0

Answer 29

skew of y= - skew of x

Question 30

kurtosis

Answer 30

kurtosis of y= kurtosis of x

Question 31

sd of y

Answer 31

|b|sd of x

Question 32

mean is

Answer 32

expected value of rv

Question 33

kurtosis is

Answer 33

proportion of outcomes in tails of distribution

Question 34

variance measure

Answer 34

dispersion

Question 35

skewness is a measure of

Answer 35

symmetry/asymmetry

Question 36

why the other three moments are central moments ?

Answer 36

because the functions involve the random variable minus its mean, X − µ. Subtracting the mean produces functions that are unaffected by the location of the mean. These moments give us information about the shape of a probability distribution around its mean

Question 37

variance

Answer 37

Var(X) = E(Xsquare) − [E(x)]square or E(X- MU)SQ

Question 38

central moments are measured relative to the mean

Answer 38

the first central moment equals zero and is, therefore, not typically used.

Question 39

Skewnes formula

Answer 39

E(x-mu)cube/ sd cube

Question 40

skewness formula explaination

Answer 40

Because we both subtract the mean and divide by standard deviation cubed, skewness is unaffected by differences in the mean or in the variance of the random variable. This allows us to compare skewness of two different distributions directly.

Question 41

kurtosis formula

Answer 41

E(X-mu)power4/ sd power4

Question 42

kurtosis

Answer 42

measure of the shape of a distribution, in particular the total probability in the tails of the distribution relative to the probability in the rest of the distribution.

Question 43

quantile function repersentation

Answer 43

Q(a)

Question 44

use of quantile function

Answer 44

A common use of quantiles is to report the results of standardized tests.

Question 45

Two quantile measures are of particular interest to us here.

Answer 45

One is the value of the quantile function for 50% called median. The second quantile measure of interest here is the interquartile range (IQR).

Question 46

IQR

Answer 46

The interquartile range is the upper and lower value of the outcomes of a random variable that include the middle 50% of its probability distribution. The lower value is Q(25%) and the upper value is Q(75%). The lower value is the value that we expect 25% of the outcomes to be less than, and the upper value is the value that we expect 75% of the values to be less than. Like standard deviation, the interquartile range is a measure of the variability of a random variable. Compared to a given distribution, the outcomes of a distribution with a lower interquartile range are more concentrated around the mean, just as they are for a distribution with a lower standard deviation.

Question 47

A linear transformation of a random variable, X, takes the form Y where Y is ?

Answer 47

Y = a + bX, where a and b are constants.