Problem 4a a random variable x is generated as follows we...

Problem 4(a) A random variable X is generated as follows. We flip a coin. With probability p, the result is Heads, and then X is generated according to a PDF fX|H which is uniform on [0,1]. With probability 1-p the result is Tails, and then X is generated according to a PDF fX|T of the form fX|T(x) = 2x, if x∈[0,1]. (The PDF is zero everywhere else.) 1. What is the (unconditional) PDF fX(x) of X? For 0≤x≤1: 2. Calculate E[X]. E[X]= Problem 4(b) We now wish to estimate the result of the coin toss, based on the value of X. 1. Find P(Tails|X=1/4). P(Tails|X=1/4)= 2. The MAP rule decides in favor of Heads if X < a. What is a?

Transcript

00:01 Let's start this problem by writing the given information, the function's giving.

00:04 The function, the probability function for heads is given to be equal to 1.

00:12 Over the interval of 0 is smaller or equal than x, which is smaller or equal than 1.

00:18 We're also given the probability function of tails to be equal to 2x.

00:24 By the way, these are 2r with respect to x.

00:27 I forgot to put the x in the parentheses.

00:30 So 2x along the same interval, because after all, our probabilities are between 0 and 1.

00:38 Then, in order to find the expected value, which is what we're being asked for, we need to do the integral from 0 to 1 of x times f of x with respect to x.

00:54 This f of x is the probability density function.

00:59 That means we need to multiply the given functions time each probability.

01:05 Heads, if the coin flips to head, is considered at success.

01:09 So for heads, the probability is p, a success.

01:14 Then for tails, we have 1 minus p.

01:20 To find the probability density function of heads is just the probability p times the function one.

01:32 Same thing, the probability, let's put h here for head.

01:35 The probability density function for tails is equal to 1 minus p, which is the probability times the function to x.

01:48 With this information, we can write our probability density function for the two of them by adding them together.

01:54 And that is just going to be that f of x, we are being asked for, which is p times 1 is 1, plus let's distribute the 2x right away.

02:05 So 2x times 1 and 2 .s times minus p...

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