Here we consider the 2n-periodic function f: R → R, that is given by f(2) = | sin()| for |æ| < . a) Use Eulers formulas to show, that the functions complex Fourier series f(x) = E-, Cneinx is given by 1 2/2(-1)" – 4 Cn = for n E Z. 16n2 – 1 -
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.
I need help with question a. I have attached a picture of the whole assignment.
![Here we consider the 2n-periodic function f: R → R, that is given by
f (x) = | sin()|
for |æ| < T.
a) Use Eulers formulas to show, that the functions complex Fourier series
f(x) = E- Cneinz
is given by
1 2/2(-1)" – 4
Cn =
for n e Z.
16n2 – 1
φ.ψε 1?(-π, π])
b) In the space L2 ([-n, T]) there is a inner product for arbitrary
by
(9, 4) = / p(x)\(x) dx.
Determine whether the Fourier series for f (x) converges uniformly on R to f (x), converges
pointwise to f (x), converges to f (x) with respect to the norm on L? ([-n, T]). Justify the
-T,
answers.
c) Her we consider the function h(x) given by the parameters 1_1, 1o and 1, as
h(x) = A-1e¬i* + do + A1e²ª.
Determine for 1-1 = -3, Xo = -2, A1 = –1
L? ([-T, 1]).
Explain how the parameters 1_1do and 1, must be selected to obtain the smallest value of the
if h has the smallest possible distance to f in
I = S", || sin(4)| –- A1ei – do – A_1e-i|² dx.
integral
Specify these values of 1_1, 2o and 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa3bc07d-af1c-413c-8f75-3b997528cb20%2F6f311337-545d-46e0-9573-285c6f35d619%2Fuqswqfa_processed.png&w=3840&q=75)
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