Chapter 5: Special Random Variables that the message will be incorrectly decoded? What independence assumptions are you making? (By majority decoding we mean that the message is decoded as "0" if there are at least three zeros in the message received and as "1" 198 Z If each voter is for Proposition A with probability .7, what is the probability that 4. Suppose that a particular trait (such as eye color or left-handedness) of a person otherwise.) exactly 7 of 10 voters are for this proposition? and is classified on the basis of one pair of genes, suppose that d represents a pure dominance, one with rr is pure recessive, and one with rd is hybrid. The pure from is ra recessive gene. Thus, a person with dd genes and dominant gene each parent. If, with respect to a particular trait, 2 hybrid parents have a toral of 4 children, what is the probability that 3 of the 4 children have the outward dominance and the hybrid are alike in appearance. Children receive 1 gene appearance of the dominant gene? 5. At least one-half of an airplane's engines are required to function in order for it to operate. If each engine independently functions with probability p, for what values of p is a 4-engine plane more likely to operate than a 2-engine plane? 6. Let X be a binomial random variable with E[X] = 7 and Var(X) = 2.1 %3D Find (a) P{X=4}; (b) P{X>12}. %3D 7. If X and Y are binomial random variables with respective parameters (n, P) ane (7, 1 – p), verify and explain the following identities: (a) P{X n– i}; (a) P{X= k} = P{Y = n– k}. %3D %3D %3D 8. If X is a binomial random variable with parameters n and p, where 0
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 5
![Chapter 5: Special Random Variables
that the message will be incorrectly decoded? What independence assumptions
are you making? (By majority decoding we mean that the message is decoded
as "0" if there are at least three zeros in the message received and as "1"
198
Z If each voter is for Proposition A with probability .7, what is the probability that
4. Suppose that a particular trait (such as eye color or left-handedness) of a person
otherwise.)
exactly 7 of 10 voters are for this proposition?
and
is classified on the basis of one pair of genes, suppose that d represents a
pure
dominance, one with rr is pure recessive, and one with rd is hybrid. The
pure
from
is
ra recessive gene. Thus, a person with dd genes
and
dominant gene
each parent. If, with respect to a particular trait, 2 hybrid parents have a toral
of 4 children, what is the probability that 3 of the 4 children have the outward
dominance and the hybrid are alike in appearance. Children receive 1 gene
appearance of the dominant gene?
5. At least one-half of an airplane's engines are required to function in order for it
to operate. If each engine independently functions with probability p, for what
values of p is a 4-engine plane more likely to operate than a 2-engine plane?
6. Let X be a binomial random variable with
E[X] = 7 and Var(X) = 2.1
%3D
Find
(a) P{X=4};
(b) P{X>12}.
%3D
7. If X and Y are binomial random variables with respective parameters (n, P) ane
(7, 1 – p), verify and explain the following identities:
(a) P{X <i} = P{Y > n– i};
(a) P{X= k} = P{Y = n– k}.
%3D
%3D
%3D
8. If X is a binomial random variable with parameters n and p, where 0 <p < 1,
show that
(a) P{X = k+1}
P n-k
P{X = k}, k= 0, 1, ..., n– 1.
%3D
1-pk+1
(b) As k
s from 0 to n, P{X = k} first increases and then decreases, reach-
goes
ing its largest value when k is the largest integer less than of e
(n+1)p.
%3D
equal to
9. Derive the moment generating function of a binomial random variable and
then use your result to verify the formulas for the mean and variance given in
the text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc75d2948-de5d-4252-902e-2fb229e41c51%2F7feb5a13-cc05-4e84-a839-1470c74f0450%2Fs1cz14q.jpeg&w=3840&q=75)
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