00:01
So to find the lower bound to the probability p absolute value x minus 5 less than or equal to 1 .5, we can use the chebyshev's inequality.
00:19
Chebyshev's inequality provides a bound on the probability of a random variable deviating from its mean by a certain amount.
00:26
It states that for a random variable x with mean u and variance standard deviation squared, and for any positive constant k, you get p of x minus the mean greater than or equal to k times standard deviation is less than or equal to 1 over k squared.
00:55
So in your case, x has a mean u of 5 and a variance, standard deviation squared, of 2.
01:08
We want to find the lower bound for p of x minus 5 less than or equal to 1 .5, which can be rewritten as p negative 1 .5 less than or equal to x minus 5 less than or equal to 1 .5.
01:31
So first let's find k.
01:35
K times standard deviation equals 1 .5.
01:40
K equals 1 .5 over standard deviation...
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