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Here to 1Select Distributionhigh probability Probability tat tne 1 pnce p1 toothpaste rootndas Sla -1highDistributions1about 0 /6sidul Il Now11...

Question

Here to 1Select Distributionhigh probability Probability tat tne 1 pnce p1 toothpaste rootndas Sla -1highDistributions1about 0 /6sidul Il Now11

here to 1 Select Distribution high probability Probability tat tne 1 pnce p1 toothpaste rootndas Sla - 1 high Distributions 1 about 0 /6 sidul Il Now 1 1



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A discrete probability distribution for a random variable $\bar{X}$ is given. Use the given distribution to find $(a)$ $P(X \geq 2)$ and $(b) E(X)$. $$\begin{array}{c|ccccc}x_{i} & -2 & -1 & 0 & 1 & 2 \\\hline p_{i} & 0.1 & 0.2 & 0.4 & 0.2 & 0.1\end{array}$$

All right. So let's say we're given a probability density function P of X on. We're told to find the value of C. That's gonna make this a probability density function over the interval. Negative. Infinity to infinity. So we can do that by saying that the integral of this probability density function over this range has equal one. So now we are. All we need to do is solve for this integral. And I might not be obvious how to do it at first. But if we multiply this integral, so integral, see negative infinity to infinity, we multiply the top and bottom, uh, top and bottom here by E to the two X. Then we get E to the X on top and then e to the two X here, plus one the ex. And so maybe that's still not clear, even, but this is actually just equal to see times the inverse tangent of E to the X. Since if you take the derivative here, you would get e to the X Times one over one plus e to the X, which is, um, which is exactly what, Uh, exactly what we have going on here. So no. Uh, now we just need to evaluate this from infinity to minus infinity. So, um, if we plug in, so see times tan, inverse tan, minus one of each of the infinity, which is infinity, minus tan, inverse of each of the minus infinity, which is zero. This is pirate too, and this is zero. So see, times pi over too. Gonna equal one. So c equals to overbuy. No, let's use that fact to ah finds the probability that X is less than or equal to four. So again it's C c is too over pie. And we already evaluated the integral which we got is the inverse 10 of e to the X And this time the bounds air just from negative infinity to minus four. So what is the sequel? So this is too over pi times the English tangent of E to the minus four of e to the minus four, uh, minus the inverse tangent of E to the minus infinity, which we already determined to zero. So your final answer is just to pi times the inverse tangent of E to the minus four, which is just equal 2.1166 approximately. Um, you can put that injury calculated to check it out. Yeah, that's it.

Yeah. The probability that the random variable X. is greater than or equal to two echoes. The probability that it equals Hamburg plus the probability that it tickles sell them Can Mexico's .018 plus .002, Which equals .02. And the expectation. Because the some of the value of exciting times its probability when she calls Miners .1 Times Point 98 it's 100 Times .018 Because 1000 times .002 can their sickles straight points 702 Mhm. Mhm.

Yeah. The probability that a random variable Is greater than or equal to two. Because the probability is that 8, 8 holes too, Since the random variable can take -2 -101 M2. And so their sickles point to. Mhm And its expectation he calls assam mhm of the value of the outside times is probability. Richie calls -2 times point to plus minus one times point to Plus zero times point to Because one times point to Plus two times point to. And this it goes miners point fel Miners .2 plus .2 plus .4 and The final answer is zero.

Mhm. The probability that the random variable Is greater than or equal to two Equals the probability that it equals two Plus the probability that it was three plus the probability that it calls for and they're psychos point to first point to plus point to Rich People's .6. Be careful and its expectation. He calls the son of the value of the outside times. Its probability, Which he goes one times .4 plus joe times point to Plus three times point to Plus four times point to okay. And this equals 0.4 plus 0.4 plus 0.6 plus 0.8 where she goes two points to


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