A pair of dice is rolled n times.
Find the probability that “seven” will not show at all.
A pair of dice is rolled n times.
Find the probability that “seven” will show at least once.
A pair of dice is rolled n times.
Find the probability of obtaining double six at least once.
Consider the coin experiment where the probability of “head” equals p
and the probability of “tail” equals 1 − p . If we toss the coin till a head
appears for the first time, what is the probability that the number of
required tosses is odd?
Determine the probability that a measurement of x will be greater
than the mean of x .
Find the variance of x .
An estimate of a parameter is said to be unbiased if the expected value
of the estimate equals the parameter. Show that the maximum
likelihood estimate of μ is unbiased.
A particular disease is known to be found in men over 65 with probability
20%. A blood test has been used to detect this disease with a 6% false
negative (i.e., the test incorrectly gives a negative result) rate and a 3%
false positive (i.e., the test incorrectly gives a positive result) rate. Note
that the positive result means the disease is found in the test, while the
negative result means the disease is not found in the test.
What is the probability that a man over 65 receives a positive test
result?
A particular disease is known to be found in men over 65 with probability
20%. A blood test has been used to detect this disease with a 6% false
negative (i.e., the test incorrectly gives a negative result) rate and a 3%
false positive (i.e., the test incorrectly gives a positive result) rate. Note
that the positive result means the disease is found in the test, while the
negative result means the disease is not found in the test.
If a 70-year-old man took the test and received a positive result, what is
the probability that he really has this disease?
The probability that a certain kind of electronic device is defective is 0.1.
An inspector randomly picks 20 items from a shipment of this type of
electronic device. Assume each test of a randomly selected item is a
Bernoulli trial. Let a random variable X denote the total number of
defective items in these 20 items.
Write down the probability distribution of the random variable X .
The probability that a certain kind of electronic device is defective is 0.1.
An inspector randomly picks 20 items from a shipment of this type of
electronic device. Assume each test of a randomly selected item is a
Bernoulli trial. Let a random variable X denote the total number of
defective items in these 20 items.
Compute the probability P(X < 2) .
The probability that a certain kind of electronic device is defective is 0.1.
An inspector randomly picks 20 items from a shipment of this type of
electronic device. Assume each test of a randomly selected item is a
Bernoulli trial. Let a random variable X denote the total number of
defective items in these 20 items.
Derive the moment generating function for the random variable X .
Show your derivation.
A random sample of size 16 from the normal distribution with the mean μ
and variance 25 yielded the estimated mean μ = 73.8 . Find a 95%
confidence interval for μ .
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