Transcript Introduction to Probability

Onur DOĞAN
Onur DOĞAN
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Suppose that a random number
produces real numbers that are
distributed between 0 and 100.
generator
uniformly
Determine the probability density function of a
random number (X) generated.
 Find the probability that a random number (X)
generated is between 10 and 90.
 Calculate the mean and variance of X.
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The number of customers who come to a donut
store follows a Poisson process with a mean of 5
customers every 10 minutes.
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Determine the probability density function of the time (X;
unit: min.) until the next customer arrives.
Find the probability that there are no customers for at least 2
minutes by using the corresponding exponential and Poisson
distributions.
How much time passes, until the next customer arrival
Find the variance?
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The standard normal random variable (denoted as Z)
is a normal random variable with mean µ= 0 and
variance Var(X) = 1.
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P(0 ≤ Z ≤ 1,24) =
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P(-1,5 ≤ Z ≤ 0) =
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P(Z > 0,35)=
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P(Z ≤ 2,15)=
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P(0,73 ≤ Z ≤ 1,64)=
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P(-0,5 ≤ Z ≤ 0,75) =
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Find a value of Z, say, z , such that P(Z ≤ z)=0,99
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A debitor pays back his debt with the avarage
45 days and variance is 100 days. Find the
probability of a person’s paying back his debt;
Between 43 and 47 days
Less then 42 days.
More then 49 days.
The binomial distribution B(n,p) approximates
to the normal distribution with E(x)= np and
Var(X)= np(1 - p) if np > 5 and n(l -p) > 5
Suppose that X is a binomial random variable with
n = 100 and p = 0.1.
Find the probability P(X≤15) based on the
corresponding
binomial
distribution
and
approximate normal distribution. Is the normal
approximation reasonable?
The normal approximation is applicable to a Poisson if λ > 5
Accordingly, when normal approximation is applicable, the
probability of a Poisson random variable X with µ=λ and
Var(X)= λ can be determined by using the standard normal
random variable
Suppose that X has a Poisson distribution with λ= 10. Find
the probability P(X≤15) based on the corresponding
Poisson distribution and approximate normal distribution.
Is the normal approximation reasonable?
Recall that the binomial approximation is applicable to a
hypergeometric if the sample size n is relatively small to
the population size N, i.e., to n/N < 0.1.
Consequently, the normal approximation can be applied to
the hypergeometric distribution with p =K/N (K: number of
successes in N) if n/N < 0.1, np > 5. and n(1 - p) > 5.
Suppose that X has a hypergeometric distribution with N =
1,000, K = 100, and n = 100. Find the probability P(X≤15)
based on the corresponding hypergeometric distribution
and approximate normal distribution. Is the normal
approximation reasonable?
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For a product daily avarege sales are 36 and
standard deviation is 9. (The sales have normal
distribution)
Whats the probability of the sales will be less
then 12 for a day?
The probability of non carrying cost
(stoksuzluk maliyeti) to be maximum 10%,
How many products should be stocked?