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The mode of & noral distribution is 65.5 and thc standard deviation is 8.4. Thcn what is thc median ofthe distribution?0 a 655b.7.79c 57,1d.73.9What is the esti...

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The mode of & noral distribution is 65.5 and thc standard deviation is 8.4. Thcn what is thc median ofthe distribution?0 a 655b.7.79c 57,1d.73.9What is the estimated value of Y when X = 10 for the regression equation1026r ?b. 138d.36The probability distribution function of random variable is givcn in the following table: T=r P4u) 045 0.25 0.2 0.25 0.15Calculate P( < T < T)

The mode of & noral distribution is 65.5 and thc standard deviation is 8.4. Thcn what is thc median of the distribution? 0 a 655 b.7.79 c 57,1 d.73.9 What is the estimated value of Y when X = 10 for the regression equation 102 6r ? b. 138 d.36 The probability distribution function of random variable is givcn in the following table: T=r P4u) 045 0.25 0.2 0.25 0.15 Calculate P( < T < T)



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A discrete random variable $X$ has the following probability distribution: $$ \begin{array}{c|ccccc} \mathrm{m} & 77 & 78 & 70 & 80 & 81 \\ \hline \boldsymbol{P}(\mathrm{w}) & \mathrm{p} .15 & 0.15 & 0.20 & 0.40 & 0.10 \end{array} $$ Compute each of the following quantities. a. $P(80)$ b. $P(X>80)$. c. $P(X \leq 80)$. d. The mean $\mu$ of $\underline{X}$. e. The variance $a^{2}$ of $X$. f. The standard deviation $\sigma$ of $X$.

Okay, So for this question, were given a densely function f of tea which is equal to 0.5 e to the negative 0.5 tea. And so we can notice that this follows exactly the format of an exponential distribution that gives us all of these formulas down here to work with. So if we want to find the mean for tea, that's just going to be expectation of tea, which we see is one over Lambda. So we're gonna get one over 10.5, which would give us 20. Next, we're gonna be looking for the standard deviation, which is the same as the square root of the variance. So we're again gonna get one over Lambda and again get 20 for the final part. We want to know the probability that, ah t is between its mean and one standard deviation above the mean. So this is between 20 and 40 so we would write this as cap f of 40 minus cap f of 20. So all we have to do is go ahead and plug 40 and 20 into our half of tea so we would get one minus you to the negative Ah, 0.5 times 40 which would be to. And then we're gonna subtract off one minus e to the negative one. And if you plug this into a calculator, you will get that. The answer is 0.232 five. So this is our final answer.

Hi, everyone. Today we'll be covering exercises from Chapter 11 1 tree named the Special Probability Density functions. So hopefully you're already comfortable with Chapter 11 12 and 11 Do in Chapter 11 country, where you take a look at true different. Ah, very useful and using everyday life probably the identity functions such as uniform ah, probability, density function. And that's the one that I could use for this problem from no one. So let's have a quick review of what is a uniform? Um probability density function. So, see, this is a graph here. Um, the if the problem is uniform, it means that it doesn't change in value. So here we have a certain So are you X in this problem, it could be the length of belief. And why is the probably I'm just gonna write product between two values of X. So between two different lengths, which we caught A and B, the probability is equal. So that's what it's telling us that between can be and so that music as likely to get the length of this length or a leaf of this length, all possibilities are equal. And for this specific case, um the function that describes this. Um, it's probability his f of X. It has to be a constant of x ray. There can't be any X in here because this is constant. You look in the manual, this is what you see. So as we consider is no ex. Well, one thing we should mention is that X has to be has to be between A and B. So why this constant? This is because the area off your probability density function has to be one. Why? Because the area of a program, the density function, is also probability. It's the probability that your experiment will be between the boundaries of the area. So let me rephrase that if you take for example, um the area I'm drawing ready here, this should be equal to the probability that the leaf is between a and this value here. And so if you think the whole area that probably as to be one has to be one, because belief has to be somewhere. So that didn't help you understand why we need this constant here. Why is it won over B minus a. Well, if you would take the integral of this rectangle. You would get the height, times, length and the length here. What the length is equal to be minus a right if you take B minus a times one over B minus A. You get one. All right, so now that's that was just a quick review of the uniformed density function. No, it's focusing that problem. Problem number one. The length in centimeter of the leaf of a certain plant is continuous. Random variable would probably density function defined by of picks because 5/7. And that tells you x between, um, four and 5.4. So what you wish you should The information you should expect from this is that in our special case of the uniform on, bro the density function, this is a and this is being, and we've just covered that it's extra vehicle to one over B minus a right. Um, if you take 5.4 minus four and you do one over to result, you get 5/7. So this is all consistent with the definition that we had previously. Now, for this exercise, we want a you know that mean the distribution. Well, since we have a special case that we've still you already. We can look in the manual and find an equation for the mean of a uniform density function, and the mean is simply a plus. B divided bite you. So that will be four was 5.4, divided by two, and this is equal to 4.7. Um, those are centimeters because we're talking about the length of leaf. Now be we want to know the standard effusion, and this one is equal to one over its credit of 12 B minus C. Again, that's an equation that you find in the in this chapter for the special case. This is only for the uniform, probably the density function. It's only gonna work when you have square box like that that is defined between a and B and that the eyes the height is one over, you might say. So it's calculated this If we put um, 5.4 minus four, that's 1.4. So 1.4 divided by square root of 12 that is approximately zero point 404 centimeters right and not see, we want to know the probability that they're random. Variable is between the mean and one standard deviation above me. All right, I'm gonna You knew that for this so that we can visualize this better. So are a function is like that, right? And then we've got hey here and would be here on We can even write that. This is for this is 5.4. We found the mean here at 4.7. This makes sense because it's right in the mill. So on average, that's the lanky yet no and B. We also found that the standard deviation, which is how likely you are to deviate from the mean to be 0.4. So that news this length here, zero point four for us. Now we want to know the probability that the leaf is between the mean and one standard deviation way from demeanor are above the mean. So we wanted to know what's the probably That belief falls into this red square here that it is between four points and centimeters and five point Wong or four centimeters. Um, all right, so the equation we need for that is, and we need to find the area of this Red Square to find the area the best way to do it is to use an interval. So 4.7 times and the indigo is from 4.7 24 27 plus zero point 44 And what we're gonna integrate is if affects that probably function. That's how you get probably that something is between two different guns. All right, so, um, let's keep going. If X is, um five, we're seven. And so we can be right. This is for integral from 4.72 5.14 of five or seven X. And if you saw this, you get you can take the constant way, and then it's simply into North Constant. It is X, but we have to be evaluated from 4.7 two 5.24. So, um, we're gonna replace X here with this number and that minus X with this number, and that gives this. I see your point two. He ate six. So that's a pro, the probably decimal, with one being 100%. So if you want to convert this into person, did you can always too times 100% and then you would get 28 point 28 points. 86 percent. All right, so that's it for that problem. You can always refer back to this problem when you have. Ah, when you want to know more about the uniform. Uh oh. The special case of the uniform probability density function, cause I'm not gonna do the review everything, um but yeah, that's a problem for you in the next one.

All right. Random variable X is normally distributed with me in new equals 10 and segment equals two. We wish to use the normal distribution to find the following probabilities A three. This question is challenging and understanding of normal distributions. So remember that a Z table maps the scores under probabilities for the standard normal distribution. So focusing first on the standard normal. Before we dive into how to utilize for variable X. We have the probability that the grid and equals peanut corresponding to the area and purple peanut to the right of ours. Not score as an example is probably greater than 0.5. We can solve for such problems using the symmetry of the normal curve and the fact that the total area under normal covers one. Specifically with itself For this problem, we need to convert our variable X to a Z score using the conversion Z equals x minus mu over sigma doing this. We can utilize in normal table or the normal distribution table with the standard normal. So Z equals 13 minutes. 10/2 is equal to 1.5. That's probably the less than 1.5 is equivalent to the probability X less than 13.9332. We proceed to solve following the same for part B Z is negative 0.5 plugging in X. That's probably the greater the negative 0.5 point 6915 and cr z scores are negative two and two. Probably easy fall between these cores is 20.9545. FD Z scores are negative four negative three. The probability is 30.13. Z falls between. Finally, an E Z scores are negative six negative one. The probability falls between these as 10.1587.

In this problem. Random variable access Normally distributed with a mean value, music was five and a sigma equals four. For standard deviation. We wish to use the normal distribution table to find the following probabilities A three week. The question is challenging your understanding of normal distributions and how to find probabilities under a curve. To start, let's review relevant information through the normal distribution. We can use this table to have the scores on the probabilities, that sort of standard normal distribution. So probably the greater than zero equals peanut means the area of peanut and purple is the area to the right of Xena under normal curve. So as example of probably does is greater than 0.5 because the standard normal with mean zero half of the curve is to the right, that's 00.5. This relates to the two principles are going to be using to solve for each of these probabilities, firstly, the symmetry, the normal curve and secondly, the fact that the total area under the normal curve is one. Since the information up here relates to how to find probabilities for standard normal curve, we need to utilize the conversion to convert our values in these probabilities to Z scores or Z is x minus mu over to think about. So for p x less than 11, this is equivalent to for z equals 11 minus 5/4 and 1.5 probably is is less than 1.5 which is from a table nine point point 9332 identically. We follow this process for b through E for br Z score is negative 1.25. So there probably is is greater. The negative 1.25 is 8.29 point four per CR Z scores are negative 40.5 point five. Probably falls between the 6.38 to 9. NPR scores are negative 1.751. The probability falls between these as .080 and three finally meet. Have Z scores of negative .75.75. There probably is, he calls between these 2.5467.


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