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# PROBABILITY DISTRIBUTIONS

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## Shashi M

on 19 August 2018

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#### Transcript of PROBABILITY DISTRIBUTIONS

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)
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