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The CDF of a random variable is given as From the CDF,I feel that the random variable here is continuous.So shouldn't the P(X=x) equals 0 here and we only need to find P(X>=1).

Aman
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    The random variable is not strictly continuous; examine that cumulative distribution function closely, let us say for $x=0$, and you should be able to see that. – jbowman Sep 08 '23 at 18:59
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    Graph $F$ and look for any vertical jumps. – whuber Sep 08 '23 at 19:20
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    Your feelings are not a reliable indicator. Draw the cdf and then look at it. – Glen_b Sep 08 '23 at 22:57

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Notice that $\lim_{x\,\uparrow\,0} F(x)=0$ and $F(0) = \frac14.$

Thus you have a jump at $0.$ The c.d.f. is not continuous at $0.$

Likewise $\lim_{x\,\uparrow\,2} F(x)= \frac34$ and $F(2)=1,$ so there is also a jump at that point. The c.d.f. is not continuous at $2.$

whuber
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Michael Hardy
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