**PROBABILITY DISTRIBUTIONS**

DEFINITIONS

Random variable:

rule enabling us to assign numbers to each outcomes of a sample space

2 voters in the 2015 election are interviewed successively- each are asked if they voted for SAJ.

S={YN, NY, YY, NN}

outcomes: YN NY YY NN

# of yes responses: 1 1 2 0

Continuous random variables:

can take values within an interval

mass, height, weight, time, temperature

amount of something

Discrete random variables

: separated or isolated points

number of children, number of tosses until a tail is obtained, number of defective products

count

Probability distributions of a random variable

: what probability is associated to the values of the random variables

DISCRETE PROBABILITY DISTRIBUTIONS

Discrete probability distribution

: specification of probabilities associated with the various distinct values of a discrete random variable

P(x)

: probability associated with value x

Consider a student taking part in a true/false quiz consisting of 2 questions. Since he has not studied, the student guesses the answers. Let x=the number of correct guesses.

S={CW, WC, CC, WW}

outcome CW WC CC WW

x 1 1 2 0

P(0)=p(ww)=1/4

P(1)=p(cw) or p(wc) = 1/2

P(2)=p(cc)=1/4

probability distribution

x 0 1 2

P(x) 1/4 1/2 1/4

BINOMIAL DISTRIBUTION

A binomial distribution must have the following characteristics:

it consists of performing an experiment a fixed number of times, n. Each time the experiment is performed, we call it a trial.

Each trial has only 2 possible outcomes: a success, S, or a failure, F.

The probability of a success is p, and the probability of failure is q (1-p). These probabilities are the same for all the trials.

The trials are independent of one another

Consider a die being rolled 3 times. Each time we record whether the value is an odd number or an even number. Let x=number of times that an even number is recorded.

S={OOO, E00, OEO, OOE, OEE, EOE, EEO, EEE}

x 0 1 2 3

P(x) 1/8 3/8 3/8 1/8

BINOMIAL DISTRIBUTION QUESTIONS

**DISCRETE PROBABILITY & BINOMIAL DISTRIBUTIONS**

P(x) is a probability distribution if

P(x) lies between 0 and 1

The sum of P(x) is 1

MEAN AND VARIANCE

Find the:

probability that x=0 or x=1

mean

standard deviation

A dentist has determined that the number of patients x to be treated in an hour is described by the probability distribution given below. Find the mean, variance and standard deviation

x P(x)

1 1/10

2 4/10

3 4/10

4 1/10

Binomial distribution formula:

Assume that when a hunter shoots a deer, the probability of hitting the deer is 0.6. Find the probability that the hunter

will hit 4 out of the next 5 deers that he shoots.

will hit at least 4 out of the next 5 deers that he shoots.

will hit at least one of the next 5 deers that he shoots.

will hit at most 3 deers out of the next 5 deers that he shoots.

MEAN AND VARIANCE

For a Bernoulli distribution, the mean and the variance are given by:

Assume that the probability of a boy being born is 0.5. If a couple plans of having 6 children, find the probability that

exactly 1/2 are boys

all are boys

all are boys or girls

there is at least 1 boy

find the mean and the variance of the distribution

85% of dishwashers manufactured by a large company do not need repair for 2 years. If 5 dishwashers are selected at random, find the probability that

all 5 will not need repairs for 2 years

at least 3 will need repairs for 2 years

It is possible for a computer to pick up erroneous signals that does not show up as an error on the screen. The error is called a silent error. A computer is defective and it introduces a silent error with a probability of 0.1. This computer is used 20 times during the week. Find the probability that

no silent error occurs

at least one silent error occurs

Would it be usual for more than 4 such errors to occur? Explain your reasoning based on probability involved

EXPECTED VALUE

For a random variable x, the expected value of x is the mean of the random variable

E(c)=c

E(cX)=cE(X)

E(cX+Y)=cE(X)+E(Y)

Let X and Y be random variables with E(X)=7 and E(Y)=-5. Compute E(4X-2Y+6)