Stats unit 5 - probability distributions

Solving expected value problems

1. Define random variables

2. Create a probability distribution table

3. Check to make sure its a probability distribution (check requirements)

4. Find expected values with formula

process to solve binomial problems

1. Define random variables

2. List and check requirements

3. Make a probability statement

4. Write binomial formula and show inputs

5. Solve

Mean of a probability distribution

E(x)=Σxipi

STD of probability distribution

√Σ(xi-μ)²P(xi)

Variance of a probability distribution

Σ(xi-μ)²P(xi)

usual values max and min

𝑥̄+or-2𝑠

unusually high probabilities

P of x or more successes is less than .05

Unusually low probabilities

Probability of x or less is less than 0.05

requirements for a probability distribution

1. 0≤P(xi)≤1

2. ΣP(xi)=1

Probability distribution problems format

1. define random variable

2. check requirements

3. write necessary formulas and inputs

4. solve

Binomial distribution requirements

title

1. fixed number of trials (n=_)

2. trials are independent

3. each trial has 2 outcomes

4. probability of success never changes (p=_)

x~B(n,p)

calculating probabilities when a random variable is binomially distributed

1. define random variable

2. check requirements for binomial distribution

3. make a probability statement

4. write formula and inputs

5. solve

plugging binomial into calc

(n, p, x)

random variable of binomial

x=number of

mean of binomial

𝜇x=𝑛𝑝

standard deviation of binomial

𝜎=√𝑛𝑝𝑞

variance of binomial

𝜎=𝑛𝑝𝑞

Calculating mean, std, variance, usual values of binomial

1. define random variables

2. list and check requirements

3. write down formula and inputs

4. solve

𝜇_x+y

E(x)+E(y)

𝜇_x-y

E(x)-E(y)

𝜇_a+x or 𝜇_a-x

a + E(x) or a - E(x)

𝜇_bx

bE(x)

var(a+x)

var(x)

var(bx)

b²var(x)

var(x+y)

var(x)+var(y)

var(x-y)

var(x)+var(y)

requirements for geometric distribution

1. NOT a fixed number of trials (until)

2. trials independent

3. 2 outcomes

4. p of success never changes

x~G(p)

Geometric probability formula

P(x=k)=pq^k-1

Geometric in calc

(p, x), 

cdf

or equal to

pdf

equal to

process to solve geometric problems

1. define random variables

2. list and check requirements

3. make a probability statement

4. write the geometric formula and show inputs

5. solve

Geometric mean

𝜇_x=(1/p)

Geometric standard deviation

√(q/p)

Geometric variance

(q/p)

Geometric defining random variables

“number of ____ until ____”