Khan Academy: Random Variables

Question 1

How are Bernoulli distribution mean and variance formulas calculated?

Answer 1

Mean=P
Variance=P(1-P)
P= chance of success

Question 2

How are Binomial distribution mean and variance formulas calculated?

Answer 2

Binomial distributions are n Bernoulli distributions, therefore:
Mean=nP
Variance=nP(1-P)
P= chance of success

Question 3

Why is variance of binomial distribution is n * variance of bernoulli distribution

Answer 3

We know for independent variables we have:
var (x+y)=var(x)+var(y)
for binomial variables, they are the sum of n bernoulli distributions, therefore, var(binomial)= n* variance bernoulli or np(1-p)

Question 4

What’s the difference between a geometric random variable and a binomial random variable?

Answer 4

no fixed number of trials

Their indicator can be: doing something UNTIL something happens
## Footnote

Ref

Question 5

How can we calculate cumulative probability for geometric random variables?

Answer 5

P(X<10) is equal to 1-P( no success in the first 9 trials.

P(X≥10) is equal to P(no success in the first 9 trials)

for cumulative probability for less than a value
for cumulative probability for more than a value

Question 6

What’s the expected value and SD of a geometric random variable?

Answer 6

E(X)=1/P
SD=√(1-P)/P
P= probability of success

Ref

Question 7

Diego is at the batting cage where they are currently offering batting cage customers their choice of The Slow Ball Challenge or The Fast Ball Challenge.
The Slow Ball Challenge: There will be 10 pitches, each at 60 mph. Diego estimates that he will hit each individual pitch %95 of the time. If Diego can hit all 10 pitches, he will win a total of $25; otherwise he will lose $5.
The Fast Ball Challenge: There will be 3 pitches, each at 90 mph. Diego estimates that he will hit each individual pitch %60 of the time. If Diego can hit all 3 pitches, he will win a total of $60; otherwise he will lose $10.
What are Diego’s expected winnings from playing The Slow Ball Challenge? Round your answer to the nearest dollar.

What are Diego’s expected winnings from playing The Fast Ball Challenge? Round your answer to the nearest dollar.

Answer 7

The probability of Diego winning The Slow Ball Challenge is 0.95^10≈.6

Now we can write our expected winnings equation for The Slow Ball Challenge.
(0.6⋅$25)+(0.4−$5)≈$13

The probability of Diego winning The Fast Ball Challenge is 0.60^3 =0.2160
Now we can write our expected winnings equation for The Fast Ball Challenge.
(0.216⋅$60) + (.784⋅−$10)≈$5

Question 8

What’s a random variable?

Answer 8

A variable whose values are obtained from a random process, like picking cards from a deck

Question 9

Carlos is playing a dice game with Elena.
Carlos will roll 2 six-sided dice, and if the sum is greater than 7, he will win $5. If the sum is 7, they will tie and he will break even. If the sum is less than 7, he will lose $4.
In rolling 2 dice, does the order matter?
What is Carlos’s expected value of playing this game? Round your answer to the nearest cent.

Answer 9

The order of numbers matters in rolling multiple dice
15/36 * $5 + 6/36 * $0+ 15/36* $4= $ 0.42

Question 10

At one store,%20 of the pairs of shoes sold are of men’s athletic shoes. Lionel rings up 8 orders which contain one or more pairs of shoes. Let N be the number of orders Lionel rings up that include at least one pair of men’s athletic shoes.
What type of variable is N?
Binomial
Geometric
Neither

Answer 10

Neither
Binomial is incorrect because: Probability of an order containing men’s shoes does not stay constant
Geometric is incorrect because: there’s a fixed number of trials.

Note that the orders are not just one pair of shoes, they’re 8 packs of shoes, in each we may have 0,1 or multiple shoes, therefore, the P(success) is not constant in each order

Question 11

Some nations require their students to pass an exam before earning their primary school degrees or diplomas. A certain nation gives students an exam whose scores are normally distributed with a mean of 41 points and a standard deviation of 9 points.
Suppose we select 2 of these testers at random, and define the random variable D as the difference between their scores. We can assume that their scores are independent.
Find the probability that their scores are within 10 points of each other.
You may round your answer to two decimal places.
P(∣D∣<10)

Answer 11

we have 2 random variables, A and B, coming from 2 normal distributions, meanA=meanB=41 and sdA=sdB=9
now we are creating a random variable D=A-B
meanD= meanA - meanB=0
sdD= radical(varA+varB)= 12.728

we want P(-10<D<10)
so it’ll be P(z= 10/12.728) - P(z= -10/12.728)= 0.57