Probability - Continuous Random Variables

Question 1

what is the probability that X takes a certain value (P(X=x))

Answer 1

zero

Question 2

what is (p.d.f)

Answer 2

the probability density function

Question 3

what is the probability density function

Answer 3

a non-negative function such that P(x <= X < x + Δx) = f(x) Δx, the area of the region enclosed by y=f(x) between x and Δx

Question 4

what is P(-∞ < X < + ∞)

Answer 4

1

Question 5

what is P(X=a)

Answer 5

0

Question 6

what is (c.d.f) F(X)

Answer 6

the cumulative distribution function

Question 7

what is the cumulative distribution function

Answer 7

the probability that X takes a value smaller than X, F(X)=P(X<=x)

Question 8

how do you find P(a<X<b) using F(X)

Answer 8

F(b)-F(a)

Question 9

how do you find P(X>a) using F(X)

Answer 9

1-P(X<a) = 1-F(a)

Question 10

what are the similarities and differences in E(X) and Var(X) between continuous and discrete r.v’s

Answer 10

the expectation for a continuous variable is the area under the curve xf(x) but variance is the same

Question 11

what does X~U(a,b) mean

Answer 11

X is a random variable with uniform distribution on the interval [a,b]

Question 12

what is the p.d.f if X~U(a,b)

Answer 12

f(x) = 1/b-a if a<=x<=b, otherwise 0

Question 13

what is the c.d.f if X~U(a,b)

Answer 13

0 if x<a, x-a/b-a if a<=x<b and 1 if b<=x

Question 14

what are E(X) and Var(X) if X~U(a,b)

Answer 14

E(X) = a+b/2
Var(X) = (b-a)^2/12

Question 15

what does X~N(μ, σ^2) mean, what are μ and σ^2

Answer 15

X is a normally distributed random variable
μ is the expected value
σ^2 is the variance

Question 16

what are some of the defining features of the normal p.d.f

Answer 16

bell shape, symmetric about μ, unimodal, standard deviation σ

Question 17

how much of the area is within one and two σ from μ

Answer 17

68% and 95%

Question 18

what does μ affect on the graph of f(x) for X~N(μ, σ^2)

Answer 18

it shifts the peak along the x axis

Question 19

what is the effect of σ on the graph of f(x) for X~N(μ, σ^2)

Answer 19

the smaller the σ, the higher the peak

Question 20

how do you calculate probabilities for normal r.v’s

Answer 20

work with the standard normal random variable

Question 21

what is the standard normal random variable of the standard normal distribution Z~N(0,1)

Answer 21

Z=X-μ/σ

Question 22

for X~N(μ, σ^2) how do you find P(a<=X<=b)

Answer 22

P(a-μ/σ<=Z<=b-μ/σ)

Question 22

what is Φ(z)

Answer 22

the probability density function for the standard normal P(Z<=z)

Question 23

what is Φ(z)=P(Z<=z)

Answer 23

the cumulative distribution function for the standard normal

Question 24

what is equivalent to Φ(-z)

Answer 24

1-Φ(z) = P(Z<=-z)

Question 25

how do you calculate P(a<Z<b)

Answer 25

Φ(b)-Φ(a)

Question 26

If X ~ N(μx , σx^2) and Y ~ N(μy , σy^2) are independent, what is the distribution of aX+b

Answer 26

N(aμx+b, a^2σx^2)

Question 27

If X ~ N(μx , σx^2) and Y ~ N(μy , σy^2) are independent, what is the distribution of X+Y

Answer 27

N(μx+μy, σx^2 +σy^2)

Question 28

what is the sample mean

Answer 28

a new random variable when there are n independent r.v’s with the same mean and variance

Question 29

what is the expectation of the sample mean

Answer 29

the mean shared by all the independent r.v’s

Question 30

what is the variance of the sample mean

Answer 30

the variance shared by all the independent r.v’s squared divided by the number of independent r.v’s

Question 31

if the r.v’s contributing to the sample mean are independent and normally distributed then what is the distribution of the sample mean

Answer 31

Xbar~N(μ,σ^2/n)

Question 32

what happens to the variance of the sample mean when the sample is large

Answer 32

it tends to zero
the mean of a large random sample from a variable is likely to be a good estimate for the expectation of the variable

Question 33

what is xp

Answer 33

the p-quantile of the distribution

Question 34

what is p when xp is the upper quartile of X

Answer 34

0.75

Question 34

what is p when xp is the lower quartile of X

Answer 34

0.25

Question 35

what is p when xp is the median of X

Answer 35

0.5

Question 36

if α is greater than 0 but less than 1, what is xα where P(X>=xα) = α

Answer 36

the critical value corresponding to α

Question 37

what is another way of thinking of xα

Answer 37

the 1-α quantile