4. Suppose that one word is to be selected at random from the sentence THE GIRL PUT ON HER BEAUTIFUL RED HAT. If X denotes the number of letters in the word that is selected, what is the value of E(X)?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
question 4
![下午6:27 12月6日 週日
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Summary
The expectation, expected
distribution. If the probab
along the real line, then the
random variable X can be
finding the distribution of -
X can be calculated directl
Exercises
1. Suppose that X has the uniform distribution on the
interval [a, b]. Find the mean of X.
2. If an integer between 1 and 100 is to be chosen at
random, what is the expected value?
3. In a class of 50 students, the number of students n; of
each age i is shown in the following table:
Age i
n;
18
20
19
22
20
4
21
3
25
1
If a student is to be selected at random from the class, what
is the expected value of his age?
4. Suppose that one word is to be selected at random from
the şentence THE GIRL PUT ON HER BEAUTIFUL RED HAT. If X
denotes the number of letters in the word that is selected,
what is the value of E(X)?
5. Suppose that one letter is to be selected at random from
the 30 letters in the sentence given in Exercise 4. If Y
denotes the number of letters in the word in which the
selected letter appears, what is the value of E(Y)?
6. Suppose that a random variable X has a continuous
distribution with the p.d.f. f given in Example 4.1.6. Find
the expectation of 1/X.
7. Suppose that a random variable X has the uniform dis-
tribution on the interval [0, 1]. Show that the expectation
of 1/X is infinite.
could earn 5.8% risk-free over that one-year period. Find
the risk-neutral price for the option to buy one share.
14. Consider the situation of pricing a stock option as in
Example 4.1.14. We want to prove that a price other than
$20.19 for the option to buy one share in one year for $200
would be unfair in some way.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ceeea51-465f-4606-b567-3c32217ff55b%2F59667a2a-0a73-4a36-8c7a-38f039deb4f4%2F4er2mzg_processed.jpeg&w=3840&q=75)
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